J. Phys. Chem. 1987, 91, 1177-1184 Then, the bands of the semiconductor particle become flat, and the flat-band condition will be established. Since the electrochemical potential of photogenerated holes is still very high a t the flat-band condition, the holes more or less react with reductants faster than electrons even in the flat-band condition. Here we consider the case where further charge separation occurs from the semiconductor particle to the bulk of the solution. In such a case, true charge (electrons) further accumulate on the semiconductor particle, and, most possibly, the potential drop acrms the Helmholtz layer is changed to shiftthe band edges to the more negative, holding the bands flat. The particle potential will come into a steady state when D, = Oh. If, on the contrary, another case occurs where a strong accumulation layer is formed in the semiconductor particle by holding the band edges pinned, the value of Vp still has meaning since, in any case, Vp represents the pseudo-Fermi level of electrons of the semiconductor particle under illumination. The amount of charge on a particle at a given pH at equilibrium in the dark is given as
where r is the radius of the particle, and pH,,, is the pH value a t the point of zero charge and is equal to 5.3 for Ti02,2' 12.5 for In203,22and 4.5 for Sn02.23 Since the shift of the electrical potential by illuminaion, Vp - Vb,is caused by the excess charge accumulated on the surface of the surface of the particle, the change in electrical charge on the particle is given as Aq = -4?r$cH(
Vb -
Vp)
(16)
Consequently, the total amount of charge on a particle under illumination, q, is given by the sum of qe and Aq. The estimated value of q assuming C, = 30 pF/cm2 are also tabulated in Table 11. Since the p H of each dispersion in this experiment was 0.4, the charge of a particle in the dark is positive for each of three kinds of semiconductor particles. In 1 M HCOOH medium, the charge of T i 0 2 and Sn02 particles under illumination has the (21) Tanaka, K.;Ozaki, A. J . Cutal. 1967, 8, 1. (22) McCann, J. F.; Bockris, J. O M . J. Electrochem. SOC.1981, 128, 1719. (23) Johansen, P. G.; Suchanan, A. S.Aust. J. Chem. 1957, 10, 398.
1177
opposite sign to that in the dark. This phenomenon has been already observed for metal oxide semiconductor particles by the measurement of mobility of p a r t i c l e ~ . ~ J ~ Comparing Vpof DSP.T-1 with that of DSP.T-2, it is seen that HCOOH causes larger shift of electron energy. This observation is the result of the fact that HCOOH accelerate the accumulation of electrons on the Ti02particle because of the current-doubling ability of HCOOH.24 We also measured Vp for TiOz particles with different sizes in 1 M H C O O H (pH 0.4) media under the same condition. The results are Vp = -0.62 V for T i 0 2 particle with diameter of 0.03 pm, Vp = -0.48 V for 0.1 pm, and Vp = -0.42 V for 0.75 pm. These data indicate that a particle with smaller size has the tendency to hold more negative Vp under illumination. The detailed analysis of the size dependence of Vp is now in progress.
Summary It was verified that steady-state electron energy of dispersed semiconductor particles can be obtained directly through the measurement using the probe method. The energy of electrons on a semiconductor particle under illumination can become high enough so that the particle can supply electrons to oxidants such as water. It was revealed that the metal electrode is a good probe for this method in spite of the large hydrogen evolution current. The shift of electrical potential of a particle dispersed in a solution by illumination described above is ascribed to the change in the amount of charge on a small semiconductor particle. This phenomenon may be called electrical potential floating effect.25 This effect should be one of the critical factors which determine the photocatalytic activity of semiconductor particles. Taking such effect into consideration, it is suggested that, whenever one considers the photocatalytic activity of a small semiconductor particle dispersion, one must notice that the electron energy under illumination may differ from that in the dark. Registry No. Ti02, 13463-67-7; In203,1312-43-2; Sn02, 18282-10-5; HC104, 7601-90-3; H20, 7732-18-5; Pt, 7440-06-4; Au, 7440-57-5; Ag, 7440-22-4; W, 7440-33-7; H2, 1333-74-0. (24) Gomes, W. P.; Freund, T.; Morrison, S. R. J. Electrochem. SOC.1968, 115, 818.
(25) Aikawa, Y . ;Sukigara, M. Extended Abstracts, 5th Japan-USSR Seminar on Electrochemistry, Sapporo, Sept. 1982; p 26.
Local Compositions In Real Mixtures of Simple Molecules Ramiin G. Rubio,* Margarita G. Prolongo,+Mateo Diaz Peiia, and Juan A. R. Renuncio1 Departamento de Quimica Fisica, Facultad de Quimicas, Universidad Complutense, 28040-Madrid, Spain (Received: February 11, 1986; I n Final Form: September 3, 1986)
From experimental data on excess properties, the so-called Kirkwood-Buff integrals, G,, have been calculated for several binary mixtures of simple molecules. The tendency of the different molecules to homo- or heterocoordinationhas been discussed qualitatively in terms of the G,j integrals. For the mixtures with almost equal sized molecules the local compositions have been estimated, and some of the models frequently used in the literature have been tested and found to be inadequate. The results for mixtures of molecules which differ in size have been discussed in terms of concentration-concentration correlation functions and compared with the behavior of mixtures of hard spheres, and of Lennard-Jones spheres.
Introduction There has been a great deal of interest in the study of fluids using statistical mechanical theories, but although a significant degree of understanding has been obtained for simple monoatomic
fluids,' enormous difficulties are still present when dealing with molecular fluids. Thus perturbation theories can be applied, with a remarkable degree of success, for correlating and predicting the thermodynamic properties either of spherical polar fluids2 or
t Departamento de Quimica, E.T.S.I. Aeronauticos, Universidad Politknica, 28040-Madrid, Spain. 'Citedra de Qdmica General, Facultad Quimica, Universidad Oviedo, Oviedo, Spain.
(1) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic: London, 1976. ( 2 ) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Oxford University: London, 1984.
0022-365418712091-1177$01 SO10
0 1987 American Chemical Society
1178 The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 nonspherical nonpolar fluid^,^ but very little is known about nonspherical polar fluids. Much attention has recently been paid to the problem of local compositions in mixtures, its relationship to the excess thermodynamic properties of the mixtures, and its dependence on the shape, size, and intermolecular potential of the m o l e c ~ l e s . ~ 'Of special interest has been the influence of the local compositions in the development of mixing rules for the characteristic parameters of the equations of state with sound statistical mechanical basis8 In the past years most of the local composition models used in the literature had an empirical character9,10or were based on the quasi-chemical hypothesis."*'* They have usually been tested by studying their abilities in correlating and predicting excess properties of multicomponent mixtures.'j Recently, Nakanishi et aL4 and Hoheisel and KohlerS obtained the local compositions for mixtures of Lennard-Jones spheres using Monte Carlo simulations; thus local-composition models can be tested in a more rigorous way. Results have shown that nonrandomness can be accounted for by the strictly regular solution model," and what is more important, perturbation theory does explain the deviations from the bulk composition within the error of the simulation^,^-^ which means that, for those mixtures, nonrandomness is mainly due to packing effects. Similar conclusions are arrived at from the computer simulations of square-well spheres.14 However, no computer simulation data are available for mixtures containing more complex components with anisotropic shapes and/or interaction potentials, so that they mimic real systems. One should expect that the larger the anisotropy in the two previous factors, the larger the nonrandomness in the solution. Furthermore, although some simulations are available for mixtures near the two-phase region$J5 the reported local compositions show some disagreement in the two-phase region; thus it remains unknown if packing effects do explain the nonrandomness in the solution. The same happens for mixtures formed by a subcritical and a supercritical fluid. Local mole fractions can be calculated from the radial distribution functions (rdf) gu, since the number of j molecules around an i molecule within a spherical volume of radius Lo is given by
4rNj
Rubio et al. has been included on the rdf in eq 1, recent work by Lebowitz et a1.18-20indicates that using adequate angle-averaged potentials one can obtain spherical effective potentials which adequately represent anisotropic potentials, as in the case of CQ2.21
Kirkwood-Buff Integrals and Local Compositions Kirkwood and BufP2 developed an exact theory of solutions which leads to the following expressions for the isothermal compressibility and the partial molar volumes in a mixture of c componentsz3 IBI
vi=
(2)
(3)
?:
where lBlij is the cofactor of Bij in the determinant lBl, and
Bij = pXi(6jj
+ XjGjj)
(4)
Another useful relationship isz3
where 6 , is the Kronecker's delta and pli means that all the chemical potentials except p j are kept constant. The theory relates the thermodynamic properties to the so-called Kirkwood-Buff integrals G,, defined such as the product
picij= p j J m 4 x G [ g j ( r )- 11 dr gives the average excess of j molecules around a central i molecule over the bulk average. The integral in eq 6 can be split
G, =
0
70
-4r2r d r
+ i 7 [ g i j ( r )- 1 ] 4 d dr +
N . . = - JLtJgij(r)r2 d r "
u
So far there is no rigorous way of obtaining gij(r)for complex molecules from statistical mechanical principles, which makes the theoretical calculation of the local composition a very difficult task. Perturbation theories allow one to calculate the thermodynamic properties using the rdf of a reference system with only repulsive forces. This makes them unsuitable for studying fluids in which attractive forces might have a significant influence on the structure of the fluid, as could be the case for multipolar'6 or hydrogen-bonded fluids." Even though no angular dependence (3) Fischer, J.; Lago, S. J . Chem. Phys. 1983, 78, 5750. (4) Nakanishi, K.; Okazaki, S.; Ikari, K.; Higuchi, T.; Tanaka, H. J. Chem. Phys. 1982, 76,629. (5) Hoheisel, C.; Kohler, F. Fluid Phase Equilib. 1984, 16, 13. (6) Lee, L. L.; Chung, T. H.; Starling, K. E. Fluid Phase Equilib. 1983, 12, 105. (7) Mansoori, G. A.; Ely, J. F. Fluid Phase Equilib. 1985, 22, 253. (8) Mansoori, G. A. In Equations of State. Theory and Applications; Chao, K. C., Robinson, R. L., Eds.; ACS Symp. Ser. No. 300; American Chemical Society: Washington, DC, 1986. (9) Wilson, G. M. J. Am. Chem. Soc. 1964, 86, 127. (10) CBnovas, A.; Rubio, R. G.; Renuncio, J. A. R. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 783. (11) Guggenheim, E. A. Mixtures; Clarendon: Oxford, 1952. (12) Panayiotou, C.; Vera, J. M. Can. J . Chem. Eng. 1981, 59, 501. (13) Kehiaian, H. V.; Grolier, J.-P. E.; Benson, G. C. J. Chim.Phys. 1978, 75, 1031. (14) Lee, K. H.; Lombardo, M.; Sandler, S.I. Fluid Phase Equilib. 1985, 21, 177. Sandler, S.I.; Lee, K.-H.; Kim, H. In Equarions ofSrare. Theories and Applications; Chao, K. C., Robinson, R. L., Us.; ACS Symp. Ser. No. 300; American Chemical Society: Washington, p C , 1986. (15) Schoen, M.; Hoheisel, C. Mol. Phys. 1984, 53, 1367. (16) Monson, P.; Steele, W. A.; Streett, W. B. J . Chem. Phys. 1983, 78, 4126. (17) Jorgensen, W. L. J. Chem. Phys. 1982, 77, 5757.
where yij is an effective distance of closest approach between molecules i and j such that g&) = 0, r < yu, and Lijis the smallest distance beyond which
The first integral in eq 7 is a volume which should be almost independent of the state of the liquid ( p , T), except perhaps for water or very highly hydrogen-bonded Ben NaimZ5 assumed that the main contribution to G, would come from the area under the first peak of g&), but this has been shown to be too drastic an approximation for highly ordered fluids like watereZ6 Nakanishi et al.4 and Hoheisel and Kohler5 have assumed L, to be the distance at which the first peak of the rdf decays in mixtures of equal-sized Lennard-Jones (LJ) spheres, which is approximately 1.5 times the size parameter of the LJ potential, aij A similar criterion has recently been suggested by Mansoori and E ~ Y . ~ (18) Lebowitz, J. L.; Percus, J. K. J . Chem. Phys. 1983, 79, 443. (19) Williams, G. 0.; Lebowitz, J. L.; Percus, J. K. J . Chem. Phys. 1984, 81, 2070. (20) Williams, G. 0.; Lebowitz, J. L.; Percus, J. K. J. Phys. Chem. 1984, 88, 6488. (21) Johnson, J. D.; Shaw, M.S. J. Chem. Phys. 1985, 83, 1271. (22) Kirkwood, J. C.; Buff, F. J . Chem. Phys. 1951, 19, 774. (23) Ben Naim, A. Waferand Aqueous Solutions; Plenum: New York, 1974. (24) Patil, K. J. J . Solution Chem. 1981, 10, 315. (25) Ben Naim, A. J. Chem. Phys. 1977, 67,4884. (26) Donkersloot, M. C. A. J. Solurion Chem. 1979, 8, 293. (27) Hanky, H. J. M.; Evans, D. J. Int. J . Thermophys. 1981, 2, 1.
The Journal of Physical Chemistry, Vol. 91, No. 5, 1987
Real Mixtures of Simple Molecules Hanley and Evans have shown that for dense soft spheres the statistical mechanical energy and virial equations reproduce the computer simulation results when the upper limit of integration is changed from infinity to roughly 2.5 times the molecular diameter. However, the simulation results of Hoheisel and Kohler5 show that for mixtures of Lennard-Jones spheres of different diameters the local compositions oscillate around the bulk composition as a function of L,, for L, < 3 u l l ,u l l corresponding to the smallest sphere in the mixture. This makes it rather difficult to determine a value of L, for which eq 8 will hold. On the other hand the local compositions are monotonous functions of L , for equal-sized molecules, thus making it possible to find an L,, value for which eq 8 is satisfied. Therefore for the rest of this section we will refer to mixtures of almost equal-sized molecules. On the other hand, Hoheisel and Kohler5 have discussed the advantages of using an unique L, value for all the i-j pairs in the solution. Thus, we will assume that taking
L , = L = 5 X max (uil,uJJ)
(9)
Equation 8 is closely satisfied.’ Anyway it should be pointed out that even though a change in the choice of LiJwould modify the absolute values of the local compositions calculated below, the larger the L, the smaller the differences with respect to the bulk compositions, and both the qualitative behavior and the test of the local-composition models would remain unaffected. The local mole fractions can be written asz8
XjpGij
1179
variables, which is in agreement with the experimental behavior of P K ~ R T .Using ~ ’ such a corresponding states principle, Mathias and O’Connell have developed correlations for systems containing a supercritical fluid.)’ Combining eq 2-6 and 12-1 5 it is easy to find relationships between the G, and the Cij integrals.
Cji G,
p - pjcij = - ( p . G . . J
JJ
p,c..,c.. JI
P J + 1) = + ___ picjj + 1 p - pjc, iJpl
(17)
Unfortunately the corresponding states correlations for Cjjhave been formulated by averaging data on a large number of components of very different chemical nature. Although this seems not to be an important problem when calculating bulk thermodynamic properties, the method leads to very poor results when predicting the G, integrals. This is not unexpected since the G,15 depend on the structure of the fluid well beyond the maximum of the first peak of the rdf. Therefore a different approach seems to be necessary, and it will be described in the next section. Calculation of the Cij Integrals from Experimental Data
The G , integrals can be calculated from thermodynamic properties after inversion of the Kirkwood-Buff t h e ~ r y ,which ~~,~~ leads to
Gjj(i#j) = Gij = RTKT- vivj DV
+ Ni:
x., IJ = Nij” + Ni: + p(XjGij+ XiCij)
(10)
where K T and V are the isothermal compressibility and the molar volume, respectively, of a mixture of mole fraction Xij, is the partial molar volume of component i , and D is defined by
where
is the bulk average of j molecules in a spherical volume of radius Ljj, and may be written as Nj: = 4rpXjLi:/3. Using eq 10 one can calculate the local mole fractions for nonpolar and polar binary mixtures formed by simple fluids using their G, values. The main difficulty is the lack of knowledge about g,(r) for realistic models of anisotropic molecules. O’ConnellZ9has developed a matrix formulation which expresses the thermodynamic properties of the fluids as functions of integrals of the direct correlation function, which is equivalent to the Kirkwood-Buff theory and leads to
p i being the chemical potential of component i in the mixture. Thus in order to calculate the G, integrals the following data are necessary: excess volumes, p,isothermal compressibility, K T , and chemical potentials, p, the latter being obtained from excess Gibbs energy data, GE, for the systems indicated in Table I. The and GE data have been fitted to a PadC apexperimental p r ~ x i m a n t from , ~ ~ which the partial molar volumes Vi and the chemical potentials have been derived by differentiation. Iso(pwTRZ9-l = 1 - ( ~ i ~ + c i2i~ i p z C i z+ P ~ ~ C ~ (12) J / P ~ thermal compressibility data are very scarce for the systems studied here. The K T data are only used to calculate the first term on the righ-hand side of eq 18. It is easy to show that this term including KT represents only a few percent of the total value of G, when the mixture is far from the critical point.32 In general a corresponding while a linear states calculation has given K T for the pure interpolation
with C, = p J o c i j ( r ) 4ar2 dr
(16)
c,(r) being the direct correlation function for the i-j pair. Among the advantages of this formulation is that the c,(r) functions are of much shorter range than gij(r);thus modelling the Cijintegrals should be easier than modelling the G, ones. Besides, Gubbins and O ’ C ~ n n e l have l ~ ~ shown that perturbation theory suggests that Cij= C;, the latter being the corresponding integral for the isotropic fluid. Furthermore, they found that, for densities p > p&j” should be a universal function when expressed as reduced (28) Rubio, R. G.; Prolongo, M. G.; Cabrerizo, H.; Diaz Peila, M.; Renuncio, J. A. R. Fluid Phase Equilib. 1986, 26, 1. (29) O’Connell, J. P. Mol. Phys. 1971, 20, 27. (30) Gubbins, K. E.; OConnell, J. P. J . Chem. Phys. 1974, 60, 3449.
has been used for the mixture, 4i being the volume fraction of its compressibility. Since eq 18 and 19 component i , and ~r(’) include partial molar magnitudes, it is very important to use precise fits of accurate experimental data to calculate the derivatives. A procedure based on the maximum likelihood principle33was used to obtain the parameters of the Pad6 approximant which fit and GE. That procedure allows one to obtain the variance-covariance matrix of the parameters, from which it is possible to estimate the uncertainties in Vi and in the factor D, eq As discussed by Matteoli and L e p ~ r this i ~ ~is a very important step since usually the uncertainties of the Gij integrals are large. A more convenient procedure would be to estimate D from light scattering experiment^^^ which directly provide ( a p i / a X J , , ;un(31) Mathias, P. M.; O’Connell, J. P. Chem. Eng. Sri. 1981, 36, 1123. (32) Matteoli, E.; Lepori, L. J . Chem. Phys. 1984, 80, 2856. (33) Rubio, R. G.; Renuncio, J. A. R.; Diaz Pefia, M. Fluid Phase Equilib. 1983, 12, 217.
(34) Brelvi, S. W.; O’Connell, J. P. AIChE J . 1965, 11, 886.
Rubio et al.
1180 The Journal of P h y s i c a l C h e m i s t r y , Vol. 91, No. 5, 1987
-100
-
0
1
0
X1
Figure 1. Gij factors (in cm3.mol-') for systems containing argon:
(-.-a)
TABLE I: References of the Experimental Excess Functions Necessary for the Calculation of the Kirkwood-Buff Integrals
system
+ Kr Ar + N, Ar
Ar Ar
+ 0, + C2H6
+ C2H4 + CH4 + C2H6 + Xe Xe + C,H4 Xe + C2H6 Xe + HCI Kr Kr Kr Kr
Xe Xe
+ HBr
+ CF4 0 2 + N2 CO + N2 CH4 + CZH, CH4 + C2H6 C2H6 + C2H4 Kr + N O
TIK
ref
115.77 83.82
a
83.82 90.69 115.76 11 5.76 1 15.76 161.38 161.39 161.39 195.42 195.42 159.01 83.82 83.82 103.94 103.99 161.39 115.76
b b c d e C
f g
e h i j
h b k c
I n2
aChui, C.-H.; Canfield, F. B. Trans. Faraday SOC.1971, 67, 2933. bPool, R. A. H.; Saville, G.; Herrington, T. M.; Shields, B. D. C.: Staveley, L. A. K. Trans. Faraday SOC.1962,58, 1692. CCalado,J. C. G.; Gomez de Azevedo, E. J. S . ; Soares, V. A. M. Chem. Eng. Commun. 1980, 5, 149. dCalado, J. C. G.; Nunes da Ponte, M.; Soares, V. A. M.; Staveley, L. A. K. J . Chem. Thermodyn. 1978, 10, 35. 'Calado, J. C. G.; Staveley, L. A. K. Trans. Faraday SOC.1971, 67, 1261. JCalado, J . C. G.; Staveley, L. A. K. Trans. Faraday SOC.1971, 67, 289. gCalado, J. C. G.; Soares, V. A. M. J . Chem. Thermodyn. 1971, 9, 911. hCalado, J. C. G.; Kozdon, A. F.; Morris, P. J.; Nunes da Ponte, M.; Staveley, L. A, K.; Woolf, L. A. J . Chem. SOC.,Faraday Trans I , 1975, 71, 1372. 'Calado, J . C. G.; Gray, C. G.; Gubbins, K. E.; Palavra, A. M. F.; Soares, V. A. M.; Staveley, L. A. K.; Twu, CH. J . Chem. SOC.,Faraday Trans. I 1978, 7 4 , 893. 'Lobo, L . Q.; Staveley, D. W.; Clancy, P.; Gubbins, K. E.; Gray, C. G. J . Chem. SOC., Faraday Trans. 2, 1981, 77, 425. kCalado,J. C. G.; Soares, V. A . M. J . Chem. SOC.,Faraday Trans. I , 1977, 73, 1271. 'Calado, J . C. G.; Gomez de Azevedo, E J. S . ; Clancy, P.; Gubbins, K. E. J . Chem. Soc., Faraday Trans. 1 1983, 79, 2657. mCalado,J. C. G.; Staveley, L. A. K. Fluid Phase Equilib. 1979, 3, 153.
fortunately these data are not available for mixtures of simple molecules. Results and Discussion K i r k w o o d - B u f f Integrals. Since the G, integrals give some
qualitative insight into the solution, we will discuss them before (35) Kato, T.: Fujiyama, T.; Nomura, H. BUN. Chem. SOC.Jpn. 1982, 55, 3368.
Ar
+ Kr;
(-.e-)
Ar
+ N2; (-)
Ar
1
X1
+ 0,; ( - - - ) Ar + C2H6.
studying the local mole fractions. Figure 1 shows the G,, for several systems containing argon. The bars indicate the estimated uncertainties in different composition regions calculated as indicated in the previous section. Subscript 1 stands for argon. The argon oxygen system presents a nearly ideal behavior according to Ben Naim's criterion2s
+
(22) G12 = (GI1 + G22)/2 The G I 2 and G22 values for the argon nitrogen and argon oxygen systems do not differ significantly. Nevertheless the values of GI, are quite different, indicating that the tendency of the argon molecules to gather is bigger in the system with nitrogen. Even though the system with krypton behaves similarly to that with nitrogen, there are some differences in the shape of the GI, and G22 curves. The tendency of nitrogen to gather increases with the argon concentration, whereas for krypton it is almost concentration-independent. The oxygen + nitrogen and nitrogen + carbon monoxide systems behave like the argon oxygen; thus we have not included them in Figure 1. A completely different situation is found for the argon + ethane system at 91 K, for which both Gll and GI2have maxima while G L 2has a sharp minimum. This indicates a strong tendency of each type of molecules to stay away from molecules of the other type. This behavior suggests a pattern similar to the two-phase separation, at least in a preliminary stage. From their vapor-liquid equilibrium experiments, Eckert and P r a u ~ n i t have z ~ ~ predicted the existence of a UCST at 81 K for this system which supports the behavior shown in Figure 1. Figure 2 shows the results for systems containing krypton as component 1. There are not many differences among the systems containing xenon, methane, or ethane, although the data for the system with xenon were measured at a different temperature. The polarity of the nitric oxide seems to result in a very different behavior, increasing the tendency of krypton to gather, especially in the krypton low concentration region. The system with ethene presents an intermediate behavior. In spite of the importance of the quadrupole-induced dipole interactions pointed out by Calado et al.,37the similarity of the quadrupolar moments of ethane and ethene makes it difficult to explain the differences in G,, in their mixtures with krypton on this basis. The G,j factors for systems containing xenon are shown in Figure 3 vs. the mole fraction of xenon. The system with carbon tetrafluoride presents sharp minima in the GI1and G22curves, as does the system with hydrogen bromide for the G22curve. These behaviors are similar to that of the argon + ethane system discussed previously. Even though most of the systems containing
+
+
+
(36) Eckert, C. A,; Prausnitz, J. M. AIChE J . 1965, / I , 886. (37) Calado, J. C. G.; Gray, C. G.; Gubbins, K. E.; Palavra, A. M . F.; Soares, V. A. M.; Staveley, L. A. K.; Twu, L.-H. J . Chem. SOC.,Faraday Trans. I 1918, 74, 893.
The Journal of Physical Chemistry, Vol. 91, No. 5, 1987
Real Mixtures of Simple Molecules
7
G11
1G12
80
1181
I -
-301
-
-
LO
0
-40
Xl
1
0
Xl
Figure 2. G , factors (in cm3.mol-') for systems containing krypton: Kr + CH,. I
C
(-.-a)
Kr
1
+ NO; (-..-)
0
1
Xl
+ CZH,; (-+-+)
Kr
+ Xe; (-)
+ CF4; ( - - - ) Xe + HBr; (-*-.)
Xe
+ HCI;
Kr
Kr
+ C2H6; (---):
1
(312 -2oc
-400
-600
-8ot
X1
1
0
Figure 3. G , factors (in cm3.mol-') for systems containing xenon: (-)
xenon present a clear nonideal behavior according to Ben Naim's criterion, perturbation theory has been able to explain the experimental values of V@ and GE for these system^,^',^^ the predictions for the systems with hydrogen chloride and bromide being of similar quality in spite of the differences depicted in Figure 3. This would be in accordance with the conclusions of Hoheisel and Kohlers and of F i ~ c h e according r~~ to whom nonrandomness would be mostly due to repulsive forces. For mixtures of Lennard-Jones spheres, Hoheisel and Kohler found that the strictly regular solution model describes relatively well the local composition results; thus a test of some of the available local composition hypothesis, using eq 6 and the GI]values obtained for real systems, seems to be useful. Mixtures of Almost Equal-Sized Molecules: Test of Some Local-Composition Models. Most of the local-composition models used in the literature are either e m p i r i ~ a l or ~ ~based ' ~ on the quasi-chemical h y p o t h e ~ i s . ' ~ -The ~ ~ most frequently used expression is9+' xlJ/x!l
= xJ/xI
exp(-'IJ/T)
1
X1
(23)
where el, is a constant characteristic of the difference in the interactions of i-i and i-j pairs. Equation 23 is used in different (38) Lobo, L. Q.; McClure, B. W.; Staveley, L. A. K.; Clancy, P.; Gubbins, K.; Gray, C . G. J . Chem. SOC.,Faraday Trans. 2 1981, 77, 425. (39) Fischer, J. Fluid Phase Equilib. 1978, 2, 91. (40) Maurer, G.; Prausnitz, J. M. Fluid Phase Equilib. 1978, 2, 91.
Xe
(-sa-)
Xe
+ C2H6.
Gibbs energy models like NRTL,41UNIQUAC," and in several group-contribution methods like ASOG42 and UNIFAC.43 ) be comAccording to eq 23 the ratio Qij = ( X j , X i ) / ( X j J j must position-independent. Panayiotou and VeraI2 have given an analytical expression for the local composition as a solution of the quasi-chemical equations. In terms of the mole fractions it can be written as 2x, with
K = 1 - exp(At/RT)
(25) A€ being a parameter which characterizes the difference of interactions between the i-i, j-j, and i-j pairs. Equation 24 leads to a Qij ratio which is composition-dependent in contrast to eq 23. Finally the strictly regular solution model" leads to (26) X,/Xjj = ( X j / X j ) [ l - ( A ~ / K r ) x j ]
which has been shown to give better results than eq 23 for mixture of LJ spheres. (41) Renon, H.; Prausnitz, J. M. AIChE J . 1968, 14, 135. (42) Kojima, K.; Tochigi, K. Prediction of Vapor-Liquid Equilibria by the ASOG Method Elsevier: Amsterdam, 1979. (43) Fredenslund, A,; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria using UNIFAC; Elsevier: Amsterdam, 1911.
1182 The Journal of Physical Chemistry, Vol. 91, No. 5, 1987
II
I
I
I
Rubio et al. 1.3
I
L
I
1 2 t
-4- 1
1.:
I-
x- x-
1.1
...-
@---
@
-...-"'
...-...-...-...-.. .._..-. ------..---
,,_
-
C..C_..C...-..._.
L
0.9
0.9
0.e
0
0.2
06
OL
08
X1
Figure 4. Nonrandom factors for systems of almost equal-sized mole-
I
I
I
I
1
02
0.4
0.6
0.8
1
cules. Curves labeled 0 correspond to Q2], and those labeled 0 to QI2. (1) Ar + 0,; (2) CO + N,; (3) Kr + CH4; (4) C2H4+ Xe; ( 5 ) C,H, + C,H,. X , is the mole fraction of the first component.
Figure 5. Nonrandom factors for systems of almost equal-sized molecules according to two models: (-) from the Gij integrals; ( - - - ) from the
TABLE 11: Lennard-Jones Size Parameters Used in This Work
quasi-chemical hypothesis; as in Figure 4.
substance Ar 0 2
NZ Kr
co Xe NO2
UlA 3.406' 3.38gb 3.602* 3.633' 3.643b 3.969d 3.775'
substance CH4 C2H6 C2H4
CF4 HC1 HBr
UIA 3.737f 4.2799 4.1408 4.25Id 3.64Ih 3.790h
"McDonald, I. R.; Singer, K. Mol. Phys. 1972, 23, 25. b H ~Y.; , Ludecke, D.; Prausnitz, J. M. Fluid Phase Equilib. 1984, 17, 217. 'Barreiros, S . F.; Calado, J. C. G.; Clancy, P.; Nunes da Ponte, M.; Streett, W. B. J . Phys. Chem. 1982, 86, 1722. dLobo, L. Q.; McClure, D. W.; Staveley, L. A. K.; Clancy, P.; Gubbins, K. E.; Gray, C. G. J . Chem. SOC.,Faraday Trans. 2 1981, 77,425. 'Lobo, L. Q.; Staveley, L. A. K.; Clancy, P.; Gubbins, K. E.; Machado, J. R. S . J . Chem. SOC., Faraday Trans. 2 1983, 79, 1399. fBarreiros, S . F.; Calado, J. C. G.; Nunes da Ponte, M.; Streett, W. B. J . Chem. SOC.,Faraday Trans. 1 1983, 79, 1869. ZCalado, J. C. G.; Gomez de Azevedo, E. J. S . ; Clancy, P.; Gubbins, K. E. J . Chem. Soc., Faraday Trans. 1 1983, 79, 2657. 'Calado, J. C. G.; Gray, C. G.; Gubbins, K. E.; Palavra, A. M. F.; Soares, V. A. M.; Staveley, L. A. K.; Twu, Ch-H. J . Chem. SOC.,
X1
(---e)
from strictly regular model. Labels
The Qij ratios show a weak composition dependence for these mixtures, which is in agreement with the results of Lee et al. for square-well sphere^.^' In order to test eq 24 we have calculated the parameter At which reproduces the value of Q12at X = 0.5. From eq 24 one can obtain
A t / R T = -2 In (X12/XI1)X=o,5 (27) Figure 5 shows the predictions of eq 24 for some systems. It can be observed that it overestimates the concentration dependence of the local mole fractions ratio. Furthermore, eq 24 leads to (xjj/xjj)X=0,5 =
(1 - K)-"*
(28)
which means that (X12/Xll)X=,,5 = (X2,/X22)x=0,5, which is in clear disagreement with the ratios shown in Figure 4. The disagreement cannot be related to the choice of L in eq 9, since increasing L would lead to values of Qu closer to 1, but if Q, is larger than unity, it would not become smaller than 1 by changing L. As in eq 23, if Ae/RT 1 0, both QZland QI2must be 21, and both must be I 1 if Ae/RT I O . Althou,gh this situation can be difficult to understand within a quasi-lattice model,5 it is easy to understand Faraday Trans. 1 1978, 74, 893. if one considers that, within the rdf formalism, the coordination numbers of both components are not equal. Formulations of the Figure 4 shows the Q, ratios calculated from the G, integrals quasi-chemical hypothesis like that of Irai and Arai,48in which according to eq 10 and using the u's given in Table I1 for mixtures the At parameter is split into two different parameters, one of for which the ratio of the diameters is smaller than 1 .OS. Although them accounting for the difference of interaction of the i-i and somewhat arbitrary, calculations of the coordination number of i-j pairs and the other one for the j-j and i-j pairs, might overcome hard spheres44support the idea that for size differences so small this problem. Results of the strictly regular solution theory, eq the local compositions should be well-behaved functions of L,. 26, are very similar to those of eq 24, and they are shown in Figure It can be observed that the random distribution is quite a wrong 5 . Thus we should conclude that none of the local-composition hypothesis even for such simple systems, despite that it is frequently models discussed in this work describes correctly the Q, ratios used in the mixing rules of equations of state of noneIe~trolytes4~~~ for simple real systems. Mixtures of Molecules with Size Mismatch. As already (44) Deiters, U. K. Fluid Phase Equilib. 1982, 8, 123. Eduljee, G.H. mentioned, Hoheisel and Kohler5 have shown that in mixtures Fluid Phase Equilib. 1982,9, 41. Sandler, S . I. Fluid Phase Equilib. 1983, 12, 189. Eduljee, G.H.; McDermot, C. Fluid Phase Equilib. 1984, 18, 103. OmnOs, R. J. Phys. 1985, 46, 139. (47) Lee, K. H.; Sandler, S . I.; Monson, P. A. Znr. J . Thermophys. 1986, (45) Flory, P.J. J . Am. Chem. SOC.1965, 87, 1833. 7, 361. (46) Lacombe, R. H.; Sinchez, I. C. J. Phys. Chem. 1976, 80, 2586.
(48) Irai, Y . ;Arai, Y . Fluid Phase Equilib. 1982, 9, 201
The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 1183
Real Mixtures of Simple Molecules 3
1.0
I
--
i \,
I
0
I
U U
I I
i
I \
2
I
0.8
i I I
//
v, 0.6 0
2
1
::
v)
0.4
0 0.2
0
0.6
0.4
1D
0.8
31
Figure 6. Concentration-concentration correlation function: ( 0 )Ar + C2&; X Xe CF,;(-) ideal system; (---) hard spheres with a diameter ratio of 0.9; (-.-.) hard spheres with a diameter ratio of 0.3.
+
of molecules of different size, the definition of the local compositions is hardly meaningful, thus making it difficult to discuss the validity of any theoretical model. In order to study the influence of packing effects, and eventually that of the attractive interactions, it would be useful to define a reference system to which a random distribution could be assessed. This is a difficult task within the framework of the G, integrals. While for mixtures of spheres packing effects arise from differences in size, the G(s are strongly dependent on the absolute sizes of each molecule (through the partial molar volumes). The structure of binary mixtures has frequently been discussed in terms of the partial structure factors48
= 1+
Ev j +o[ g i j ( r ) - 11 exp(iij.7) di:
(29)
where 4 is the wavenumber. It is obvious that the long-wavelength limits of a,@) are directly related to the Gij integrals. This limit is conveniently discussed in terms of those of the number-conand Sm(q)introduced centration structure factors Sm(4),SNc(q), by Bathia and T h ~ r n t o n ~ ~ Xi2aii(ii) = x i ’ S N ~ ( i i ) + X
J N C G )
+ Scc(ii) - XiXz
xzza,z(ii) = Xz’%~(ii) - XzSNc(ii) + Scc(ii) - Xixz
xixzaiz(ii)= X~XZSNN(?) + (x2 - X P N C ( ~- )
-
SCCG)
+ XJZ
(30)
In the limit q 0 it can be shown that the concentration-concentration correlation function Scc(0) is given by S C C ( 0 ) = XJ,/D
(31)
with D given by eq 20. For the case of a mixture of equal-sized hard spheres it holds that Scc(0) = XJ,, thus being an adequate reference for the case of random distribution. Furthermore the Scc(0) values depend on the difference in size of the molecules, which is convenient as discussed previously. GeertsmaM has shown (49) Bathia, A. B.; Thornton, D. E. Phys. Rev. B 1970, 2, 3004. (50) Geertsma, W. Physicu B (Amsterdam) 1985, 132, 337.
0.2
0
0
I
I
I
I
0.2
0.4
0.6
0.6
1
*1
Figure 7. Concentration-concentrationcorrelation function for mixtures of Lennard-Jones spheres: R = O , ~ / U S~ ~=; (c1ie22)~~2/ei2;Q = t 2 2 / t i l . (1) R = 1 , s = 1, Q = 2; (2) R = 1 , s = 1, Q = 2.5; (3) R = 0.8,s= 1.3, Q = 1.7; (4) R = 1, S = 1.2, Q = 1.5.
that the Scc(0) values determine the phase diagrams of the mixtures and diverge in the vicinity of phase transitions. This will be useful since, as we have seen, the Gij factors indicate such a situation in some of the systems studied in this paper. Figure 6 shows the values of Scc(0) calculated from the Gij factors for the systems ethane argon and xenon carbon tetrafluoride; the sharp peaks confirm that the systems are not far from the liquid-liquid transition. Using the equation of state for mixtures of hard spheres5’ and eq 3 1 we have calculated the Scc(0) values for mixtures of different diameters at the packing fraction 9 = 0.49 and for ul/uz = 0.9 and 0.5. The results are shown also in Figure 6 as well as those for the ideal mixture, and they agree with those reported by Giunta et al.52 It can be observed that decreasing the ratio of the diameters from 0.9 to 0.5 prevents concentration fluctuations with respect to the ideal solution. This means that in going from 0.9 to 0.5 there is an increase in the tendency to heterocoordination due to the topological arrangements which are energetically more favorable for the hard spheres. Recent calculations of Gallego and Silberts3 for mixtures of hard ellipsoids and of hard spherocylinders of different sizes lead to similar conclusions. Thus increasing the packing effects leads to a behavior which is opposite to the one found for some of the
+
+
(51) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523. (52) Giunta, G.;Abramo, M. C.; Caccamo, C. Mol. Phys. 1985,56, 319. (53) Gallego, L. J.; Silbert, M. Chem. Phys. Lett. 1986, 125, 80. (54) Frenkel, D.; Mulder, B. M.; McTague, J. P . Mol. Crysr. Liq. Cryst. 1985, 123, 119. Frenkel, D.; Mulder, B. M. Mol. Phys. 1985, 55, 1171.
J. Phys. Chem. 1987, 91, 1184-1199
1184
systems studied here. This is somehow not an unexpected result the LB rule. Curve 3 corresponds to a mixture in which both the since mixtures of hard spheres do not show any liquid-liquid phase differences between the reduced temperatures and the departures transition. Even though mixtures of very anisotropic hard convex from the LB rule are larger than in curve 3; however, the introduction of differences in size strongly prevents concentration bodies present this kind of transitions, it it unlikely that anisotropy fluctuations, and so the tendency to homocoordination in the of shape can explain the Scc(0) values of the systems shown in Figure 6. mixture. Figures 6 and 7 show that not far from the liquid-liquid As pointed out by Ashcrofts5 immiscibility is driven by both phase transition not only packing effects but also attractive forces potential energy and entropy considerations. While the long-range are important in determining the distribution of the molecules in portions of the interaction potentials are important for the former, the solution. It remains to be known if perturbation theories the entropy is mainly determined by the correlations resulting from describe such a distribution under those conditions. It also remains the short-range interactions. In this way Joarder and S i l b e ~ t ~ ~ unclear what is the quantitative influence of nonrandomness on have shown that it is possible to obtain values of Scc(0) larger the thermodynamic properties of the mixtures. In the normal than that of the ideal mixture when the attractive part of the liquid range, far away from the phase separation, it seems that interaction potential is taken into account within a mean field packing effects account for any influence of this type. Patterson theory. Recently Kojima et calculated the G , integrals for et aL5*have recently shown that the W-shape of the CpEcurves mixtures of Lennard-Jones spheres within the Percus-Yevick of some liquid mixtures of complex molecules might be due to approximation. We have calculated the Scc(0) for some of the their proximity to an UCST. As we have seen, this situation cases studied by Kojima et al., the results being shown in Figure corresponds to sharp peaks in ScC(O),and thus to a tendency to 7 . As can be observed, increasing the difference in interaction homocoordination. Both the strong positive contributions to CpE energies between the pure components increases the concentration and to S,(O) are related to the divergences of the partial structure fluctuations above the ideal mixture for mixtures of equal-sized factors a,@) near the UCST. An analogous situation must exist spheres, assuming the Lorentz-Berthelot (LB) combination rule. in dilute solutions near the critical point of the solvent (typical Curve 4 corresponds to equal-sized spheres for which the reduced of supercritical extraction experiments), and thus noticeable temperatures of the pure components are intermediate cases bedifferences between local and bulk compositions could be expected in this case. tween curves l and 2, but there is a noticeable departure from (55) Ashcroft, N. W. In Liquid Metals, Vol. 1, Evans, R.; Greenwood, D. A,, Eds.; The Institute of Physics: Bristol, 1977; Institute of Physics Conference Series No. 30. (56) Joarder, R. N.; Silbert, M. Chem. Phys. 1985, 95, 357. (57) Kojima, K.: Kato, T.; Nomura, H . J . Solurion Chem. 1984, 13, 151.
Registry No. Ar, 7440-37-1; Kr, 7439-90-9; Xe, 7440-63-3; N, 7727-37-9; 0, 7782-44-7; C,H,, 74-84-0; NO, 10102-43-9; CIH,, 7485-1; CH,, 74-82-8; CF,, 75-73-0; HBr, 10035-10-6; HC1, 7647-01-0. ~~
(58) Patterson, D.; Costas, M.; CBceres, M., unpublished results.
Irreversible Thermodynamics of Heterogeneous Systems Byung Chan Eu Department of Chemistry, McGill University, Montreal, Quebec H3A 2K6,Canada (Received: February 19, 1986; In Final Form: June 6, 1986)
Global macroscopic evolution equations are derived for heterogeneous systems from the local field equations which were obtained previously on the basis of the kinetic theory. The global equations may be applied to systems removed far from equilibrium. Based on the theory, the equilibrium conditions are studied and the Gibbs phase rule is recovered for a reacting, heterogeneous system. Thermodynamic stability of heterogeneous systems is also analyzed. As an illustration of application of the theory, a calculation is presented in which the efficiency of an irreversible Carnot cycle is obtained. Its efficiency is one-half the efficiency of the reversible Carnot cycle: Teff = (1/2)(1 - T,/T,).
I. Introduction Macroscopic processes are often studied in a single phase system which is assumed to be so large that the boundaries do not affect the processes of interest. This is of course a simplification of the reality, but it is quite reasonable for many situations we face in nature. However, many natural phenomena occur in a condition that involves a number of phases or subsystems, either homogenous or inhomogeneous, and hence the entire system is globally heterogeneous. Examples are numerous: substances consisting of heterogeneous phases; biological systems made up with cells of various kinds that are separated by membranes and intercellular fluids and communicate with each other through chemical and physical means; mechanical engines consisting of reactors, pistons, and reservoirs, or more generally energy converters; electrical cells; an array of electronic logical units, etc. Therefore, when looked at from the standpoint of such heterogeneous systems, a single phase consideration of irreversible processes is clearly a special case of a more general theory of heterogeneous systems, which should be a global theory by necessity. It is the aim of this paper 0022-3654/87/2091-1184%01.50/0
to formulate a theory of irreversible thermodynamics of processes in heterogeneous systems and to study some of its general consequences. If the system as a whole is away from equilibrium, there necessarily exists a process in the system which arises owing to the spatial and temporal inhomogeneities in thermodynamic intensive properties, and as a result the system is generally driven to its steady or equilibrium state while at the same time energy and matter are transported within the phases (or subsystems) and across the phase boundaries. The steady state may coincide with the thermodynamic equilibrium state, but the coincidence is not generally assured since the system may tend to move away from equilibrium if there exists more steady states than one. In this work we take an approach in which global evolution equations for macroscopic variables are derived from the local continuum mechanics equations for macroscopic variables which strictly obey the thermodynamic laws. It is then possible to study the approach to equilibrium by the system and in particular the equilibrium conditions which will turn out to be the Gibbs phase 0 1987 American Chemical Society
