J. Phys. Chem. 1984, 88, 2689-2697 (w/w) naphthalene. As shown in Figure 3, at this temperature the solubility quickly reaches a limiting value at high pressures. At 39.8 OC, the solubility of solid naphthalene in supercritical xenon increases at a much faster rate than at the two previous temperatures, however, again the solubility quickly reaches a limiting value of approximately 5.4% (w/w) at high pressures. At 45.0 O C the solubility isotherm exhibits behavior which is significantly different from that of the other solubility isotherms. A large increase in naphthalene solubility occurs when the pressure is increased above only 120 atm. This large increase in naphthalene solubility is a consequence of operating very close to the mixture UCEP. At much higher pressures the solubility also reaches a limiting value, but now this limiting solubility (9.5% (w/w)) is much greater than the limiting solubility at any of the other isotherms. On the basis of solubility behavior depicted in
2689
Figure 3 the UCEP should be in the vicinity of 125 atm and 47 “C. Notice that the 45 OC isotherm represents liquid solubilities at low pressures and solid solubilities at high pressures. This solubility behavior verifies that the S-L-G line is crossed at two different pressures at this temperature. Hence, a temperature minimum must exist in the S-L-G line in the vicinity of 45 OC. As previously mentioned, the solubility of naphthalene has been determined in supercritical methane,’ ethane,* ethylene,6 and carbon On the basis of our findings, naphthalene solubility in supercritical xenon is higher on a mole fraction basis than any of the above-mentioned supercritical solvents at equivalent pressures and temperatures. We believe that this is the first time a noble gas has been shown to be a good supercritical solvent. Registry No. Xenon, 7440-63-3; naphthalene, 91-20-3.
FEATURE ARTICLE Ionic Fluids Kenneth S. Pitzer Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: December 6, 1983)
The statistical mechanics and thermodynamics of ionic fluids are reviewed and discussed on a corresponding-states basis and with emphasis on areas of recent advances. First, the vapor-liquid and critical properties are considered for pure ionic fluids such as NaCl as well as for the primitive ionic model. The underlying reasons are discussed for differences between fluids of ions and those of neutral molecules. Then, systems involving a solvent are considered with emphasis on simple systems showing phase separation or with strong ion pairing. Estimates are made for the critical curve for the system NaC1-H20 to the critical point of pure NaCl at 3900 K.
Introduction Ionic fluids have long been of interest to physical chemists. The early interest was in very dilute aqueous electrolytes. More recently, concentrated aqueous electrolytes, including mixtures, have received attention including systems continuously miscible to the fused salt. Highly charged ions introduce new features which also appear for 1:l electrolytes in solvents of low dielectric constant. Water becomes such a solvent at very high temperature. The critical properties and phase relationships of a pure salt such as NaCl represent an even more extreme case with unit dielectric constant. In this paper I will discuss several of these topics where there have been recent advances starting with the case of the critical properties of pure NaCl and of the primitive, hard-core, ionic fluid model. With certain approximations, ionic fluids should follow the principle of corresponding states. Consequently, it is useful to consider the properties of various systems on the basis of the reduced variables T I = T / T, and VI = V / V,. Figure 1 presents an overview in terms of reduced temperature and volume with molar concentration also indicated. The solid curve shows the liquid and vapor properties of pure NaCl with a critical temperature of 3900 K and volume of 490 cm3.mol-1. The phase boundaries for a molecular fluid, argon, are shown as a dashed curve for comparison. For various ionic systems in solvents the expected relationships for the critical constants are T,” = T , ’ ( Z ’ ’ / Z ~ 2 ( D ~ ’ / D ’ ‘ U ’ ’ ) 0022-3654/84/2088-2689$01.50/0
V / = V,‘(a”/a’)3 where 2 and a are the number of charges and the collision diameter, respectively, for the solute and D is the dielectric constant (relative permittivity) of the solvent. The approximate reduced temperatures are indicated in Figure 1 for simple aqueous 1:l electrolytes at 300 and 573 K and for 2:2 electrolytes at 300 K. Since for ionic systems in solvents the equivalent temperature is given by the product DT, one obtains a high reduced temperature in real systems by raising D even at the expense of some reduction in T . An aqueous 1:l system at room temperature has a very high reduced temperature. Indeed, it is impossible to obtain simple ionic fluids at such high reduced temperatures because there is dissociation yielding electrons. The resulting plasmas are interesting systems but will not be considered in this paper. The saturated vapor of a pure ionic fluid is comprised almost entirely of neutral molecules. Ion pairs predominate with substantial concentrations of dimers M,X, and some larger clusters. It is the liquid rather than the vapor where there is a major difference between the ionic fluid and a fluid of neutral molecules. One difference is the much greater thermal expansion of the ionic liquid; this is seen by comparison of the solid and dashed curves for the liquid volume in Figure 1. The ionic fluid is very highly expanded at the critical point. These characteristics will be explained in terms of the Coulombic potential with its repulsive nature between ions of the same sign. With increase in reduced temperature at vaporlike dilution, the pattern gradually changes to a mixture of ions and ion pairs and 0 1984 American Chemical Society
2690
Pitzer
The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 rnol*dni3 for NaCl
IO
0. I
I
0.0 I
---
- " I
0.5
I .o
0. I
1
IO0
IO
TABLE I: Density of NaCl Monomer and the Total for All Vaaor Saecies (as a Perfect Gas) and of Liauid NaCl
T/K 2000 2400 2800 3200 3600 4000
NaCl(g) 0.001 11 0.0043 0.0102 0.018, 0.026~ (0.035)
total(g) 0.0018 0.0071 0.017 0.031 0.046 (0.07)
TABLE II: Critical Properties of Pure NaCl" TC/K 3900 3800 TJK d,/g.cm' 0.12 0.14 pc/bar Vc/cm3.mol-' 490 pcVc/RTc 420
liquid 1.109 0.912 0.712 0.507
0.302 (0.09)
3900 258 0.39
3800 237 0.31
For alternate estimates, see Figure 3. I
( : I In
-
'"1 0.5
H20
I
I
I
a t 300 K
I Free Ions D-H Distribution (:I In H20 at 573 K
I
I
2
0
4 log
8
v,
Figure 2. The region of ion pairing for ionic fluids including the boundary for 50% ion pairs for pure NaC1.
eventually to completely dissociated ions. Figure 2 shows the curve for 50% association into ion pairs. At high concentration, however, the fused-salt pattern remains valid with change in reduced temperature. At high reduced temperatures there is complete dissociation into ions, but the behavior of the ions is still influenced by the long-range Coulombic forces in a manner given by the familiar Debye-Huckel theory. Phase Relationships for Pure NaCl Sodium chloride is an appropriate prototype ionic fluid. Kirshenbaum et al.' measured the liquid density of NaCl to high temperature and estimated the critical properties from extrapolations of these results and of the then available literature data concerning the vapor. Iz made use of a more fundamental statistical thermodynamic treatment for the vapor and revised the extrapolation to the critical point. In particular, the molecular properties of the ion-pair monomer NaCl are now known in considerable detail including the anharmonic effects. Thus, the statistically calculated properties in ref 3 are reliable to very high temperature. The dimer NazClz is also an important species, but its molecular properties are known only to the rigid rotatorharmonic oscillator approximation for the ring structure. A better estimate of the dimer population is available from the calculations of Gillan4 based on the primitive ionic model. Gillan found that open-chain structures of the dimer are important at very high temperature whereas only the ring structure is significant at the lower temperatures where the existing experimental measurements were made. It was assumed that the ratio of dimers and larger _ _ _ ~
~~~
,
0
I
6
~
(1) Kirshenbaum, A. D.; Cahill, J. A,; McGonigal, P. J.; Grosse, A. V. J . Inorg. Nucl. Chem. 1962, 24, 1287-96. (2) Pitzer, K. S. Chem. Phys. Lett. 1984, 105, 484-8. (3) Stull, D. R.; Prophet, H. Natl. Stand. Re$ Data Ser. (U. S., Natl. Bur. Stand.) 1971, NSRDS-NBS 37. (4) Gillan, M. J. Mol. Phys. 1983, 49, 421-42.
2030
2500
3000
-4-
3500
T/K
Figure 3. An extrapolation of the liquid and vapor densities of NaCl to higher temperature with alternate estimates of the critical point. The curve K is the earlier extrapolation of Kirshenbaum et al.'
-I
2.5c
B -
2.0
0 L
n ~
-0
1.5
I .o
2 .O
4.0
3.0
5.0
IO4( T/K)-'
Figure 4. The satuhted vapor pressure of pure NaCI. The circles show the alternate estimates for the critical point.
clusters to monomers from Gillan's calculations could be applied to the accurately known population of the monomeric NaCl. Gillan also considered various charged species in the vapor, but their concentrations were all very small. The resulting saturated vapor and liquid densities for pure NaCl are given in Table I and shown in Figure 1. The liquid density is based on the equation of Kirshenbaum et al. d = 2.061(1
- T/4330.7) g ~ c m - ~
(1)
reduced at very high temperature by an amount equal to the vapor density. This is the familiar assumption that the mean density
The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 2691
Feature Article is linear in temperature, the law of rectilinear diameters. In Figure 3 are shown two smooth curves extrapolating to critical temperatures of 3800 and 3900 K. The corresponding values of other critical properties are given in Table I1 while Figure 4 shows the vapor pressure. There are many uncertainties in the extrapolation of Figure 3. In the first place, there is no direct evidence for the law of rectilinear diameters for an ionic fluid. Secondly, the ratio of higher clusters to monomers for NaCl could be either higher or lower than that estimated from Gillan’s calculations for the primitive model. If there are more large clusters, the critical point could be below 3800 K by a substantial amount. However, in the range where the dimer population is known, the ratio of dimer to monomer for NaCl is somewhat less than the Gillan value. If this ratio remains smaller at very high temperature, the critical point could be a little higher than 3900 K. Thus, I regard 3900 K as the most probable value, but the uncertainty lies mostly on the side of lower temperatures. The population of ions near the critical point is small, but there is appreciable dissociation into N a and C1 atoms, possibly about 5%. The effect of this dissociation on the vapor density is less than the uncertainty concerning larger clusters; consequently, no correction was made for the population of atoms. Theory for the Primitive Model and Comparison with Experiment Before considering other real systems, it is of interest to consider the primitive, hard-core, ionic model for which the properties are now known with moderate accuracy. This model implies the potential uij = Zze2/(4ne,,Dr)
for r
>a
(2a)
ui, = +a for r < a (2b) Here the ionic charges are *Ze, D is the dielectric constant or relative permittivity, r the interionic distance, and a the hard-core diameter. The permittivity of free space is eo. However, much of the literature in this area is written in esu whereupon the 4mo factor disappears. In subsequent equations one may recover the esu form by substituting (4?r)-’ for eo. This potential fulfills the requirements for corresponding-states b e h a ~ i o r ;Le., ~ . ~the potential differs for various substances only by distance and energy scaling factors. The distance factor is clearly a. In the general case the energy scaling factor is @e2/(4m,,Da). Thus, it is convenient to define reduced variables for concentration, reciprocal temperature, and pressure as follows:
c* = ca3
(3a)
@* = @Z2e2/(4?reoDa)
(3b)
p* = pa4(4m,,D/(Z2ez))
with @ = l / k T . Also note that c is the total concentration of individual ions (both + and -) whereas the volume in Table I1 is for a mole of ion pairs. The Debye-Hiickel equation for solute properties is consistent with corresponding states. In terms of these reduced variables, the limiting-law activity coefficient is In y+ = -aI/2(@*)3/2(c*)l/2 Very recently, Gillan4 treated the ionic fluid vapor on the basis of the primitive model by calculating properties for ion pairs and larger clusters, both charged and neutral. Their respective concentrations were determined by equating the chemical potential among these species with that for the liquid. For the liquid he used Monte Carlo calculations of Lar~en’-~ with further interpretation to obtain properties for the saturation pressure. While there remain many approximations in Gillan’s calculations, all seem to be of limited magnitude. His vapor pressures and vapor and liquid densities are the only reasonably accurate theoretical (5) Pitzer, K.S. J . Chem. Phys. 1940, 7, 583-90. (6) Reiss, H.; Mayer, S. W.; Katz, J. L. J . Chem. Phys. 1961, 35, 820-6. (7) Larsen, B. Chem. Phys. Lett. 1974, 27, 47. (8) Larsen, B. J . Chem. Phys. 1976, 65,3431-8. (9) Larsen, B.; Rogde, S . A. J . Chem. Phys. 1978,68, 1309-11
003
0 04
0.05
0.06
(P*i‘
Figure 5. An extrapolation of liquid and vapor concentration for the primitive ionic model to higher temperature with alternate estimates of
the critical point. TABLE 111: Critical Properties of the Primitive Ionic Fluida A mean B PC* 17.2 17.05 16.9 CC* 0.050 0.04, 0.042 1% PC* -3.62 -3.59 -3.56
aEstimates A and B refer to curves on Figure 5. values for the primitive ionic model. While Gillan sketched extrapolations toward the critical point, I believe a presentation analogous to that of Figure 3 for NaCl can give a better estimate. For the liquid below (@*)-’= 0.045, Gillan found that the reduced concentration was accurately represented by c* = 0.951(1 - 15.56/@*) (4) This result is shown as the straight line in Figure 5. The vapor concentration becomes significant on this graph only for (@*)-’ above 0.05. In this range the liquid concentration will probably decrease below the linear extrapolation and the true curve through the critical point will be essentially like curves A and B in Figure 5. Table I11 gives the various critical properties corresponding to the two curves in Figure 5 and the mean. These results are, of course, approximate for other reasons as well as the extrapolation of Figure 5. Thus, the total uncertainty is greater than that indicated by the differences in Table 111. Earlier calculations of the critical properties for the primitive ionic fluid were presented by McQuarrie’O and by Stell et al.” Friedman and Larsen12 discussed many aspects of ionic fluids on the basis of the primitive model and corresponding states. BlanderlZbreviewed dimensional methods as applied to ionic fluids. There are also theoretical calculations for more realistic models for pure fluid NaC1,13J4but these models are much more complex and the calculations do not extend as close to the critical point as those for the primitive model. The most straightforward comparison between NaCl and the primitive model can be made for the liquid density since eq 1 and 4 are of exactly the same form when one recognizes that @*is a reciprocal temperature. Thus, one can extract from comparison of the numerical coefficients the model parameters a = 2.818 8, and Z = 1.066. The first is very close to the nearest-neighbor distance in crystalline NaCl and only a little larger than the corresponding distance of 2.6 8,reported15 for liquid NaC1. Also, (10) McQuarrie, D.A.J . Phys. Chem. 1962, 66, 1508-13. (1 1) Stell,G.;Wu,K. C.; Larsen, B. Phys. Rev. Lett. 1976, 37, 1369-72. (12) (a) Friedman, H. L.; Larsen, B. J . Chem. Phys. 1979, 70, 92-100. (b) Blander, M. Adu. Chem. Phys. 1967, 11, 83-115. (13) Lewis, J. W.E.; Singer, K. J . Chem. SOC.,Faraday Trans. 2 1975, 71, 41. (14) Lantelme, F.;Turk, P.; Quentrec, B.; Lewis, J. W. E. Mol. Phys. 1974, 28, 1537-49. (15)Edwards, F.G.;Enderby, J. E.; Howe, R. A,; Page, D. I. J . Phys. C 1975,8, 3483-90.
2692 The Journal of Physical Chemistry, Vola88, No. 13, 1984 this value of 2 does not depart very much from the true value of unity. With these model parameters, one can compare the calculated and experimental vapor densities for ion pairs. At 8* = 20 the primitive model yields 0.0035 gm~m-~ whereas one obtains from ref 3 a value of 0.022 a t the corresponding temperature of 3370 K. Thus, the experimental density or pressure is larger by a factor of 6. The data for larger clusters are much less accurate, but the values also differ substantially. Thus, it is clear that, if the model parameters are determined to fit the liquid density over a range in temperature, the vapor properties do not agree. One can, instead, fit both the critical temperature and density with the parameters a = 2.57 %I and 2 = 1.00, but now the temperature dependence disagrees for both liquid and vapor densities. The largest difference is in the heat of vaporization, which is much smaller for NaCl than for the primitive model for either of these sets of parameters. There are experimental data’ for KC1 which have been compared* both with the primitive model and with NaCl on a corresponding-states basis. For all aspects except the monomeric ion pair, KCl and NaCl follow corresponding states quite closely with the KC1 parameters T, = 3470 K and V, = 625 cm3.mol-’. The ion-pair concentration for KC1 is somewhat smaller than predicted on this basis, however. In comparison with the primitive model, the differences from NaCl and KCl are the same except for the ion pair. Since the ion pair is much less stable in the primitive model than for NaCl on this basis of comparison, the difference of the model from KCl is less but still large. Measurements in the critical region have been reported for NH4C1I6 and for both BiC1317and BiBr3.18 While the bismuth halides show substantial conductance in the liquid state, their thermodynamic properties follow the pattern for a normal molecular fluid with a large acentric factor. Ammonium chloride represents a very special case. The vapor is dissociated to NH3 + HC1 while the liquid behaves as expected for NH4+ + C1- in a manner similar to that for KCl. Since the vapor is different for NH,Cl, however, the critical properties are very different from those of KCl or of the primitive model. Coulombic Potentials and Ionic Fluid Properties The most important difference between the interparticle potentials for the ionic and the molecular fluids is the long-range repulsive potential between ions of the same sign. The long-range aspect of attractive potentials between ions of opposite sign is significant, but for many purposes these forces are screened to short range by the pattern of alternating charges. It is the repulsive Coulombic forces that lead the ionic fluid to shift with rising temperature to a low coordination number and eventually to a pattern of separated ion pairs. For molecular fluids all intermolecular interactions are attractive at longer distances and, to a first approximation, are pairwise additive. Thus, the cohesive energy, in this approximation, is proportional to the coordination number until geometrical overcrowding and extra repulsive forces arise. This overcrowding is not significant for six or even eight nearest neighbors, which are the largest coordination numbers of interest for ionic systems. Figure 6 shows the ionic lattice energy for the primitive model for a series of coordination numbers. This is just the Madelung constant.19 Linear and triangular lattices give values for C N of 2 and 3 , respectively, while both square-planar and tetrahedral lattices give about the same values for C N = 4. The value for CN = 6 is from the NaCl structure. The straight, dashed line indicates the approximate pattern for neutral molecules. Increased freedom of thermal motion can arise either from an increase in interparticle distance above that determined by re(16) Buback, M.; Franck, E. U. Ber. Bunsenges. Phys. Chem. 1972, 76, 350-4; 1973, 77, 1074-9. (17) Johnson, J. W.; Cubicciotti, D. J . Phys. Chem. 1964, 68, 2235-9. (18) Johnson, J. W.; Cubicciotti, D.; Silva, W. J. J. Phys. Chem. 1965,69, .. .. .-. 1989-92.
(19) Tosi, M. P. Solid State Phys. 1964, 16, 1-120.
Pitzer I .E
I
I
I
I
I
I .e
-
N
c1
I .c
W
a 0.5
i /
C
/
//I /.
/ I
I
I
1
I
1
2
3 C.N.
4
5
Figure 6. T h e energy of an ionic lattice as a function of the number of nearest neighbors. T h e corresponding function for neutral molecules is indicated by the dashed line.
pulsive forces or from a decrease in coordination number. From Figure 6 it is clear that the relative cost in lattice energy for a decrease in coordination number is less for the ionic than for the molecular solid and presumably similarly so for the irregular lattices of fluids. This is easy to understand. For the ionic system, the decrease in attractive energy is largely compensated by a decrease in repulsive energy of next-nearest neighbors when the coordination number decreases. There is no such compensation for the molecular fluid. The model just presented is, of course, overly simple, but inclusion of realistic short-range repulsive forces does not change the qualitative picture. The molecular dynamics calculations of Lewis and Singer13 offer support for these ideas, but the calculations do not extend to high enough temperature to give a full picture. Unfortunately, Lar~en’-~does not report the effective number of nearest neighbors for his calculations for the primitive model. It is well established, however, that the most probable nearest-neighbor distance in NaCl and KCl actually decreases with a decrease in coordination number from the solid to the liquid and from the liquid to the species in the vapor. It is this decrease in the number of nearest neighbors which yields the remarkably large rate of thermal expansion of the ionic liquid and the very large critical volume in relation to liquid volume at the triple point. These are aspects in which the ionic fluid differs remarkably from molecular fluids. The saturated vapor in both cases is “molecular”. The entropy of vaporization is somewhat less for the ionic fluid; this is reasonable in view of the open structure of ionic liquids at high temperature. But this last aspect is a rather small quantitative difference. The possible effect of polarizability of ions should also be considered. It is most important for the ion pair where there is a large field strength at an ion site. As the distribution of neighbors becomes nearly symmetrical, the field strength becomes small by cancellation, and no significant dipole is induced in an ion. Thus, polarizability is not ordinarily included in potential models for ionic crystals or liquids, but it is important for ion pairs and other small clusters in the vapor.2*22 The extra stability of the NaCl ion pair, as compared to that of the primitive model, is readily understood in terms of the ion polarizabilities and the soft repulsive forces. (20) Rittner, E. S. J . Chem. Phys. 1951,19, 1030-5. (21) Berkowitz, J. J. Chem. Phys. 1958,29,1386-94; 1960,32,1519-22. (22) (a) Brumer, P.; Karplus, M. J. Chem. Phys. 1973,58,3903-18; 1976, 64, 5165-78. (b) Secoy, C. H. J . A m . Chem. SOC.1950, 72, 3343-5. (c) Marshall, W. L.; Gill, J. S.J . Inorg. Nucl. Chem. 1974, 36, 2303-12.
Feature Article
The Journal of Physical Chemistry, Vol. 88, No. 13, I984
Ionic Fluids with Solvents Most ionic fluids involve solvents such as water with polar molecules. If the solvent retains its properties nearly constant regardless of the concentration of ionic solute, then it may be a reasonable approximation to treat the solvent as a continuous medium of dielectric constant (relative permittivity) D. The Debye-Huckel theory makes this assumption which is almost always valid for dilute solutions with large solvent DT. This assumption of a continuous dielectric cannot be valid for systems very concentrated in salt where the ions are in direct contact or are separated by only one layer of solvent molecules. Also, if a solvent is very compressible, as it is near its critical point, then one cannot assume a constant dielectric constant for dilute and more concentrated solutions. The ions will compress the solvent around and between them. There are examples of simple ionic fluids in dielectrics with reduced temperatures ranging from high values to near unity. But there are complications associated with all cases showing phase separation. Aqueous U02S04displays a pattern of phase separation rather close to that expected for a 2:2 electrolyte with a critical point near 560 K.22b This system is known, however, to be much more electrolyte; hence, the simcomplex than a simple U022+,S042ilarity of phase behavior to expectations for a simple 2:2 electrolyte may be accidental. There are several closely related systems, of which NH4FeC14 in diethyl ether is an example, where phase separation occurs at very low concentration, and even the more concentrated phase shows low conductance. Friedman23 has discussed this situation in terms of interactions of the ion-pair molecules with their very large dipole moments. A description of these systems as simple ionic solutes in a dielectric seems inappropriate. Aqueous NaCl shows phase separation at temperatures above the critical temperature of water. Near the T, of water, this phase separation is primarily related to the water which is steamlike or liquidlike in the two phases. But at higher temperatures and high pressure, 823 K and 754 bar for example, the water properties are more nearly uniform, and it is a useful but crude approximation to treat the water as a uniform dielectric with the properties of pure steam at this temperature and pressure. The detailed discussion of ionic solutions will begin with systems at high reduced temperatures and proceed to examples at lower reduced temperatures and finally to cases with phase separation and critical behavior. Ionic Solutions at High Reduced Temperature. The familiar aqueous 1:l or 2:l electrolytes near room temperature are at high reduced temperature as indicated in Figure 1. In this range the special properties arising from the ions in dilute solution are given by the Debye-Huckel theory. The net effects of various shortrange forces a t higher concentration have been described by at least four different types of expressions. First is a virial series similar to that used for nonideal gases. For electrolytes it was shown by M a ~ e that r ~ the ~ virial coefficients were functions of ionic strength as well as temperature and pressure. Semiempirical equations of the virial-series type have been s u c ~ e s s f u l ~ ~for- * ~ mixed as well as simple electrolytes and have been reviewed elsewhere.** There is an interesting electrostatic contribution to the second virial coefficient for unsymmetrical mixtures; this was described recently in this journal.29 In view of the extensive discussion elsewhere of virial expressions, no detailed account will be given here. A second type of expressions combines a Debye-Huckel term with terms of the type used for nonelectrolyte solutions. A single (23) Friedman, H. L. J . Phys. Chem. 1962, 66, 1595-1600. (24) Mayer, J. E. J. Chem. Phys. 1950, 18, 1426-36. (25) Pitzer, K. S.; Mayorga, G. J . Pfiys. Chem. 1973, 77, 2300-8; 1974, 78, 2698. (26) Pitzer, K. S.; Kim, J. J. J . Am. Chem. SOC.1974, 96, 5701-7. (27) Harvie, C. E.; Weare, J. H. Geochim. Cosmochim. Acta 1980, 44, 981-97. (28) Pitzer, K. S. In “Activity Coefficients in Electrolyte Solutions”; Pytkowicz, R. M., Ed.; CRC Press: Boca Raton, FL, 1979; Chapter 7. (29) Pitzer, K. S. J . Phys. Chem. 1983,87, 2360-4.
2693
Margules or van Laar term frequently suffices. This form of expression is especially useful for systems continuously miscible to the fused salt, and examples of this type of treatment will be given below. A third type of expression assumes one or more equilibria in which ion pairs or other associated species are formed. This type of treatment was described for the saturated vapor for the pure ionic fluid. Where such associated species have a substantial and unambiguous population, the equilibria for their formation must be considered, and this can be done in connection with expressions of either of the two types discussed above. The dissociation of HS04- in sulfuric acid solutions30 and the carbonic acid-bicarbonate-carbonate equilibria31have been treated by using virial expressions for the activity coefficients of the various species. An example will be given below where there is ion pairing in a system miscible to the fused salt. Treatments have also been presented in which all departures from a Debye-Huckel activity coefficient are ascribed to ion association even if there is no other evidence for the presence of the associated species. While such treatments usually can represent the observed properties if enough equilibria are assumed, they are computationally much more complex than either of the first two types. Also, there is a tendency to give undue credence to the calculated concentrations of various species whose existence is not independently verified. Finally, there are equations based on ionic lattice models. While such a basis may be reasonable at high concentration, it is unconvincing for dilute solutions where it fails to yield the DebyeHuckel limiting law. Consequently, I will not give further attention to this type of treatment but will list a few references for lattice and for ion association ~ a l c u l a t i o n s . ~ ~ - ~ ~ Systems Continuously Miscible to the Fused Salt. In concentrated ionic systems the long-range Coulombic forces are screened. Consequently, the short-range effect of the Coulombic forces can be treated on the same basis as the various nonionic interactions. A departure from ideal behavior arises from the difference between the attractive energy of an ion-solvent interaction and the mean of ion-ion and solvent-solvent interactions. Thus, one can write wkT = (c/2)(CMS + ‘XS - CMX - ‘SS) (5) where c is the effective coordination number and the e,, terms are respectively the attractive interaction energies for a cation M with solvent S, an anion X with solvent, the screened ion interaction MX, and the solvent-solvent interaction. Now w becomes a nonideality parameter analogous to that for nonelectrolyte solutions. Let us simplify our equations by considering only symmetrical electrolytes with mole fractions of solvent 1 and solute 2 on an ionized basis x2 = 2 n 2 / ( n l 2n2) x1 = nl/(nl 2n2) where n2 is the number of stoichiometric moles of the electrolyte M X in the solution. Then a first approximation to the Gibbs energy of mixing is A,G/RT = nl In x1 2n2 In x2 wnlxZ (7) with activities
+
+
+
a1 = XI71 and activity coefficients
a2 = (X*r*)2
+
(8)
In y1 = wx22 (9a) In y+ = w x l z These equations represent a single Margules term. There are a (30) Pitzer, K. S.; Roy, R. N.; Silvester, L. F. J . Am. Chem. Sot. 1977, 99, 4930-6. (31) Peiper, J. C.; Pitzer, K. S.J . Chem. Thermodyn. 1982, 14, 613-38. (32) Garrels, R. M.; Thompson, M. E. Am. J. Sci. 1962, 260, 57-66. (33) Garrels, R. M.; Christ, C. H. ‘Solutions, Minerals, and Equilibria”; Harper and Row: New York, 1965. (34) Pytkowicz, R. M. “Activity Coefficients in Electrolyte Solutions”; CRC Press: Boca Raton, FL, 1979.
2694
The Journal of Physical Chemistry, Vol. 88, No. 13, 1984
XI
Figure 7. The activity of water for several water-salt solutions over the full range of composition. I
I
‘
In 7
-0.05 0.6
-I .
0
0.2
2 0.4 0.6
0.8
u
0.0
1.0
Figure 8. The activity coefficients of water and of the salt over the full range of composition. The dashed lines omit the Debye-Huckel effect whereas the solid curves include it. variety of more complex equations for nonelectrolytes which are also applicable on this basis for ionic systems. Most frequently used is the van Laar equation wherein the mole fraction is replaced by a more flexible composition fraction in the single nonideality term. Figure 7 shows the solvent activity for a variety of aqueous ionic systems. The similarity of the curves to those for nonelectrolytes is evident. For adequate representation of all properties orle must also add a Debye-Huckel term. This is obviously needed on a theoretical basis. Figure 8 shows the effect of the Debye-Hiickel term for the aqueous system with the mixed salt (Li,K)N03which melts just above 373 K. The solid curves show the results when a Debye-Huckel term is included; the dashed curves are for the case when it is omitted. The difference for In y1can be seen only on the expanded scale of the inset in Figure 8 where the lower dashed curve gives a simple fit with In y1 = 0 at x1 = 1.0. One can view the curves for In y+ as arising from the integration of the Gibbs-Duhem equation from the pure fused salt (xl = 0). Up to xI 0.8 the result is unambiguous, and only a single curve is shown. Thereafter, however, the slope d In yl/dxl is slightly smaller for the true curve than is given without a Debye-Hackel term. Since this slope is multiplied by (xl/(l - x , ) ) in this integration, the effect of the small change of slope becomes very large for In y+ as x1 1.0. Thus, the Debye-Hockel effect has a large effect for the solute activity even if its effect on solvent activity is very small. The equations with van Laar and DebyeHuckel terms are given in detail in the Appendix. These aqueous systems of very large solubility have been discussed more fully elsewhere.35 Systems Showing Ion Pairing. It has now been shown that at low reduced temperature the dilute (vapor) phase of an ionic system is almost completely associated into ion pairs and larger
-
(35) Pitzer, K. S . Ber. Bunsenges. Phys. Chem. 1981, 85, 952-9.
Pitzer neutral clusters while at high reduced temperature there is complete dissociation with ion distributions given in good approximation by the Debye-Huckel theory. Figure 2 shows the curve for 50% association calculated3 for pure NaCl. This calculation is unambiguous at large molal volumes where the Debye-Huckel corrections are small. The point of 50% association at high concentration depends somewhat on the exact definitions chosen; hence, that portion of the curve is dashed to indicate some ambiguity. A familiar category of partially associated ionic systems are the aqueous 2:2 electrolytes at room temperature. As shown in Figure 2, the fraction association remains below 50%, but it is significant. There are now a number of good theoretical treatments3G38for the primitive model with parameters appropriate for aqueous 2:2 electrolytes. The grand canonical Monte Carlo method was applied by Valleau et a1.,36x37who also discuss other results. The radial distribution function for unlike ions shows an enhanced population of near neighbors, but the primitive model gives no unambiguous definition of ion pairs. Given an adjustment in the hard-core diameter, there appears to be no conflict between the experimental thermodynamic properties of 2:2 electrolytes and the model calculations. But the Monte Carlo and integral-equation methods are inconvenient for practical data representation. It is possible to devise a second virial coefficient3’ with a special ionic strength dependence that allows good representation of the data for aqueous 2:2 electrolytes near 300 K including that for mixture^^^,^^ with other salts. At higher temperatures (lower reduced temperatures) ion pairing will become stronger, and a different treatment may be needed for aqueous 2:2 systems. The same situation applies to 1:l electrolytes in solvents of low dielectric constant. The alternate procedure is to adopt some division of the population of ions around a given ion into one subpopulation to be treated by linearized equations of the Debye-Huckel type and a second subpopulation or, strictly, a probability of a closely located counterion yielding an ion pair. Then an association equilibrium is assumed to determine the population of ion pairs while the remaining population is treated by Debye-Huckel methods. Bjerrum@suggested this ion pairing method almost immediately after the 1923 discovery of Debye and Huckel. He suggested that the division be made by distance from the central ion and at the radius RB = Z 2 e 2 / ( 2 D k T ) which , is 14.4 A for an aqueous 2:2 electrolyte. He showed that this was the minimum in the radial population density of counterions at very low concentration. While the electrostatic energy is still twice the thermal energy at this radius, it is found that the equations of linearized theory give a good approximation for the more distant population as a whole. Presumably this is true because the approximation improves rapidly with increasing radius and with the screening which enters at finite concentration. Bjerrum also offered a theoretical value for the association constant K A for the “primitive model”. Since short-range forces in real systems certainly differ from this model, it is usually better to assume that the association constant itself is the parameter to be determined empirically. The dividing radius R , (rather than the hard-core diameter a) enters the equation for the activity coefficient of the unassociated ions. The Bjerrum definition is satisfactory for very dilute solutions, but at still quite low concentration one must also consider ion triplets and higher clusters. Thus, for 0.02 M and RB = 14.4 %, with a central ion pair, there is a random probability of 30% that either another ion or ion pair is present within radius RB. If this is an ion, one has an ion triplet which contributes to conductance. For thermodynamics one must consider both triplets and quadruplets. These effects have been noted experimentally for con(36) Valleau, J. P.; Cohen, L. K.; Card, D. N. J . Chem. Phys. 1980, 72, 5942-54. (37) Valleau, J. P.; Cohen, L. K. J . Chem. Phys. 1980, 72, 5935-41. (38) Rossky, P. J.; Dudowicz, J. B.; Tembe, B. L.; Friedman, H. L. J . Chem. Phys. 1980, 73, 3372-83. (39) Pitzer, K. S. J . Solution Chem. 1974, 3, 539-46. (40) Bjerrum, N. K . Dan. Vidensk. Selsk. Mat-Fys. Medd. 1926, 7, 1-48.
The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 2695
Feature Article
-+
I
I
\
L D-H
010
Distribution
15
20
R/a
Figure 9. The radial charge density for an aqueous 2:2 electrolyte. The
points are Monte Carlo calculated values. The Debye-Hiickel distribution for the same model is shown dashed if all ions are included or as the lower solid curve for a reduced population of free ions when ion pairing is assumed. ductance41 and for a c t i v i t i e ~ .At ~ ~higher ~ ~ ~ concentrations even larger clusters must be considered. Thus, the Bjerrum definition becomes inconvenient for most practical purposes. If an ion-pairing treatment is needed, an alternate definition is preferable with a different division of population^.^^ Assume that around a test ion a Debye-Huckel distribution of other ions extends down to the repulsive force cut off at an effective radius a from a central ion, and then classify as ion pairs the excess of oppositely charged ions. The formal equations for conductance and thermodynamic properties remain the same as for the Bjerrrum definition, but the repulsive radius a enters the Debye-Huckel equation for the activity coefficient of the free ions instead of the Bjerrum radius R B . We discuss below the ionic distribution on this basis. Figure 9 shows the radial charge distribution (R/a)z(g+ - g++) from Monte Carlo calculation^^^ for an aqueous 2:2 electrolyte with a = 4.2 A. Also shown as a dashed curve is the DebyeHuckel distribution, assuming full ionization for the same model electrolyte. The true population of oppositely charged near neighbors is much larger than the Debye-Huckel distribution, but these curves cross at about 7.5 A, which is only half of R B . If one ascribes an excess of near neighbors to an ion-pair population, the remaining Debye-Huckel population is given by the lower solid curve which smoothly joins the Monte Carlo results at large R values. The ion-pair population is then the difference between the solid curves, which drops essentially to zero at R / a = 2. Davies4*first used an ion-pairing definition which was stated differently but is essentially similar to the one favored here, and he found that the apparent degree of association reached a maximum and then decreased with further increase in concentration. This “redissociation” is consistent with the properties of pure ionic systems where the vapor is ion paired but the liquid is treated as fully ionized. There is no need to consider ion triplets or higher clusters on this basis. For further discussion we take an example with a nonaqueous solvent and strong ion pairing but where there is complete miscibility with a fused salt. The reduced temperature is above unity, since there is no phase separation, but it is not far above. Specifically, the system is tetra-n-butylammonium picrate in n-butanol at about 373 K where D is 9.4. The conductance and viscosity were measured by S e ~ a r d ?and ~ the vapor pressure of butanol by Pitzer and S i m o n ~ o n . The ~ ~ activity of butanol is shown in (41) Davies, C. W. “Ion Association”; Butterworths: London, 1962. (42) Gardner, A. W.; Gluekauf, E. Proc. R. SOC.London, Ser. A 1969, 313, 131-47. (43) Pitzer, K. S.J. Chem. Soc., Fafaday Trans. 2 1972, 68, 101-13. (44) Seward, R.PJ&m&Yae#%&ip Isat, 73, 515-7.
Figure 10. The activity of butanol in the system tetra-n-butylammonium picrate in n-butanol with measured values and a calculated solid curve. The dashed line indicates ideal solution behavior. 15
I
I
/
1
I
1.0
0
-2
0 log (c/rnol drn-3)
Figure 11. The total probability of finding oppositely charged ions within 12 A as a function of concentration for the picrate system. Also shown
is the ion-pair fraction 8. Figure 10. It is apparent that there is a large positive deviation from ideality. In the concentrated range the activity data can be fitted without ion pairing. An equation with van Laar and Debye-Huckel terms is adequate. But the conductance measurements clearly indicate substantial ion pairing in the very dilute range. A comprehensive treatment45includes an ion-pairing equilibrium with an association constant of 2.65 X lo4 on a mole fraction basis and fits both the activity data over the full range and the conductance data in the dilute range where their interpretation is unambiguous. The equations on a mole fraction basis for this full treatment are given in the Appendix. Although the fraction of defined ion pairs decreases in the redissociation range, the total probability of finding oppositely charged ions close to one another increases steadily with concentration. This is shown for the picrate-butanol system in Figure 11. The ion-pair fraction 0 reaches a maximum somewhat above 50% at about 0.01 M, but the probability of finding oppositely charged ions within 12 A increases steadily. Above about 0.1 M one expects to find more than one oppositely charged ion near a given ion. In the fused salt there are several near-neighbor ions of the opposite sign but no distinct ion pairs. It should also be noted that the molecular nature of real solvents often gives a natural definition for ion pairing. Thus, it is sometimes possible by sound absorption46 or s p e c t r o s ~ o p i c ~ ~ (45) Pitzer, K. S.; Simonson, J. M., submitted for publication. (46) Atkinson, G.; Petrucci, S . J . Phys. Chem. 1966,70, 3122-8.
2696
The Journal of Physical Chemistry, Vol. 88, No. 13, 1984
Pitzer
TABLE IV: Test of a Model for the Critical Curve for NaCI-H,O TJK p,/bar D, steam D, model 913 1237 4.48 4.01 923 1082 4.65 4.23 873 922 4.83 4.41 823 754 4.94 4.14 123 422 4.80 5.39
methods to distinguish between contact ion pairs and ions separated by one or more solvent molecules. Ion-pair concentrations determined by such methods can be incorporated into thermodynamic treatments if desired. Compressible Solvents: Water above T,. As a final section of this discussion of ionic fluids, consider the system NaCl-H20 above the critical temperature of water. Just above the critical temperature and pressure of pure water, the behavior of the system is dominated by the properties of water. But at considerably higher pressure, H 2 0 is only moderately compressible, and one can assume it to constitute a dielectric with constant properties. Also, the critical composition is sufficiently dilute that the ions are separated, on the average, by about two layers of water molecules. Thus, one can apply a very simple corresponding-states model. On this basis the product OCTOshould be constant, where D, is the dielectric constant of the solvent at T, and pc along the critical curve of the two-component system. For pure NaCl, D = 1 and T , = 3900 K. Table IV gives a comparison of values of D, c a l ~ u l a t e dfrom ~ ~ *the ~ ~expression 3900/T, and from an equation% for the dielectric constant of steam. From 823 to 973 K the agreement is within about lo%, which is very good for such a simple model. As the critical point of water is approached, this model cannot possibly apply, and the results for 723 K show the beginning of this deviation. A second check is the critical volume which should be the same per mole of NaCl for aqueous NaCl as for pure NaCl for this model. The experimental data for critical volumes extend only to 823 K where Urusovasl reports a density equivalent to about 480 cm3 mol-’, in remarkable agreement with the 490 cm3 mol-’ for pure NaCl (Table 11). It is known from conductance measurementss2 that there is substantial ion pairing in dilute NaCl in steam. But concentrated solutions of NaCl under these conditions appear to be fully ionized. This pattern is similar to that described in the preceding section except that phase separation is observed for the NaCl-H20 system. It is relatively easy to represent the thermodynamic properties in the fully ionized region. An equations3with a single Margules term and a Debye-Hiickel term suffices for the very wide range from 373 K to the highest temperature for which there are adequate measurements, 823 K, provided the solvent density remains reasonably high. A somewhat more complex treatments4 is required to cover the dilute region with ion pairing and give critical behavior at the correct temperature, pressure, and composition. Figure 12 shows the activity curves for H 2 0 and NaCl at 823 K and 754 bar. Supercooled liquid NaCl is taken as the reference state in this treatment; this avoids the extreme behavior of infinitely dilute NaCl near the critical point of pure H20. Since aqueous NaCl at the critical pressure and 823 K can be treated rather satisfactorily on the basis of the dielectric constant of steam under these conditions, one can estimate the critical curve above this temperature. On the basis D,T, = 3900 K, one cal(47) Davis, A. R.; Oliver, B. G. J . Phys. Chem. 1973, 77, 1315-6. (48) Urusova, M. A. Rum. J . Inorg. Chem. (Engl. Transl.) 1974, 19, 450-4. (49) Sourirajan, S.;Kennedy, G. C. Am. J. Sci. 1962, 260, 115-41. (50) Pitzer, K. S. Proc. Nail. Acad. Sci. U.S.A. 1983, 80, 4575-6. (51) Urusova, M . A. Russ. J . Inorg. Chem. (Engl. Transl.) 1975, 20, 17 17-21. . . (SGQuist, A. S.; Marshall, W. L. J . Phys. Chem. 1968, 72, 684-703. (53) Pitzer, K. S.; Li, Yi-gui Proc. Nail. Acad. Sci. U.S.A., in press. (54) Pitzer, K. S . ; Li, Yi-gui Proc. Nutl. Acad. Sci. U.S.A., in press.
1.0
0.5
0
X2
Figure 12. The activities of H,O and N a C l a t 823 K. The dashed line indicates ideal behavior on an ionized basis for a,.
30
I xx)0
I
1000
2000
4000
T/ K
Figure 13. The critical p-T curve for aqueous NaC1.
culates 0, for a given T,, then the density of steam from the dielectric constant equation,s0 and finally the pressure from the equation of state for H20.ss Both of these last equations are now being extrapolated to temperatures above the range of experimental data, but they have a sound theoretical form such that extrapolation is plausible. This estimated critical curve is shown in Figure 13. The values of the pressure above 1000 K are uncertain, however, by at least 20% and possibly more. In the calculation of the critical pressure it is assumed that any pressure effect arising from NaCl is negligible as compared to the steam pressure. Obviously, this is not true near 3900 K, but even at 3500 K the steam pressure is about 4 times the critical pressure of pure NaCl at 3900 K. Thus, in Figure 13 the curve was just drawn smoothly between the steam pressure below 3500 K and the pure NaCl pressure at 3900 K. Further refinement of this simple model is not justified at present. Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U S . Department of Energy, under Contract No. DE-AC03-76SF00098. Appendix The complete equations for a miscible 1:l electrolyte including ion pairing, a van Laar term, and a Debye-Huckel terms6are given below. These equations are readily simplified to eliminate ion pairing or to reduce the van Laar term to a Margules term; hence, the equations will not be rewritten in the simpler forms. The modification for a 2:2 or more highly charged symmetrical electrolyte is very simple. For unsymmetrical electrolytes the ( 5 5 ) Haar, L.; Gallagher, J. S.;Kell, G. S. “Proceedings of the 8th Symposium on Thermophysical Properties, 1981”; Sengers, J. V., Ed.; American Society of Mechanical Engineers: New York, Vol. 11, pp 298-302. (56) A very concise, modern presentation of Debye-Hackel theory is given in ref 36, pp 5943-4. See also ref 28, p 170 and Appendix A.
J. Phys. Chem. 1984,88, 2697-2702 equations become more complex. If the fraction of solute associated is 6, one defines mole fractions of free ions xi (either or -) of pairs x p and of solvent x, xi = (1 - 6)n2/[nl (2 - e)n,] = I, (Ala)
2697
In a2 = 2(ln (2xJ
(I,'/'
+
+ xp = enz/[nl + (2 - 6)n2]
(Alb)
xs = nl/[nl + (2 -
(Ale)
The ionic strength on a mole fraction basis I, is equal to xi in this case as indicated. The van Laar term involves two parameters. First is q which is often regarded as the volume ratio Vl/V2 for the two components but is alternatively taken as an adjustable parameter. The coefficient w of eq 5 and 9 is now generalized with w1 = qw2
(A2)
The van Laar composition variables may be written ZI
= qnl/(qnl + 2nd
(A3a)
z2
= 2n2/(4nl
+ 2nd
('43b)
where the omission of 0 implies the assumption that the nonelectrolyte effects are independent of ion pairing. The basic composition variables x1 and x2 were defined in eq 6. The ion-pairing equilibrium constant can be derived from a general equation for the Gibbs energy with the result
The solvent activity is given by
In al = In x,
+ w1zZ2+ 2A,I,3/2/(1 + PI,'/^)
(A5)
The solute activity is generalized from eq 8 to
+ w2(zlz- 1) - A , [ ( 2 / p )
+ PI,^/^) + + p I , ' / ' ) ] ) (A6)
In (1
- 2Z>/')/(I
and the basis is now the infinitely dilute reference state. The fraction association is not necessarily zero for the pure liquid on this model. Consequently, it is best to calculate activities based on the pure liquid reference state by difference from the value given by eq A6 with x2 = 1.0. The remaining quantities are the Debye-Huckel parameters on a mole fraction basis45~56 A , = (1 /3)(2?rN~dl/Ml)'/~(e~/(4rgDkT))~/~ (A7) p
= a(2e2N~dl/(Ml€,DkT))1/2
('48)
with d l and M I the solvent density and molecular mass, respectively, D the dielectric constant or relative permittivity, and e the electronic charge. If this charge is in esu, the permittivity of free space €0 should be replaced by (4?r)-*. The hard-core diameter or closest interionic distance is a. This quantity is estimated from structural information or determined empirically. If 0 is set to zero, there is no ion pairing and x, = x1 and x i = I , = x2/2. The van Laar term simplifies to a Margules term if q = 1.0 whereupon w1= w2,z1 = xl, and z2 = x2. The results for multiple-charged ions can be obtained by multiplying I , and A, each by the square of the number of units of charge. Normally K , is evaluated from conductance data with the assumption that ion pairs do not contribute and with an appropriate equation for the effect of ionic strength on ion mobility. A, is determined from solvent properties which also determine the ratio p / a . Thus, p is determined if a can be estimated from structural data. The parameters w2,q, and sometimes a are evaluated from thermodynamic data. But there are reasonable limits on q and a in view of other information. Thus, in a sense K, and w are the only freely adjustable parameters.
ARTICLES Electronic Relaxation Processes of Rare-Earth Chelates of Benzoyltrifluoroacetone Seiji Tobita, Masayuki Arakawa, and Ikuzo Tanaka* Department of Chemistry, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo, Japan (Received: February 1, 1983; In Final Form: November 7, 1983) The electronic relaxation processes in the chelates of La3+,Gd3+,Lu3+,and Na+ with benzoyltrifluoroacetone have been investigated quantitativelyby means of the measurements of quantum yield, lifetime, and transient absorption. The paramagnetic metal ion Gd3+has been found to enhance significantly both the T I So radiative (kPT= 350 s-I) and nonradiative (kGT = 66.7 s-l) rates from the ligand locally excited triplet state, whereas the corresponding effects of diamagnetic La3+,Lu3+, and Nat ions are negligibly small compared with that of the GdSt chelate. A simple perturbation treatment clearly shows that the exchange interaction between ligand ?r-electrons and unpaired 4f electrons in the Gd3+ion is responsible for the observed enhancement. Although the triplet yields @TM are determined to be as low as 0.1-0.25, all the chelates employed in the present experiment exhibit nonfluorescent behavior at room temperature. Only a certain nonradiative process, which is one of the characteristic properties of P-diketone type complexes, is found to occur except for the intersystem crossing originating from the SI state.
-
Introduction Since an original work by Weissman' and the subsequent discovery of organic chelate laser^,^-^ a number of studies have (1) Weissman, S.I. J. Chem. Phys. 1942, 10, 214. (2) Whan, R. E.; Crosby, G. A. J . Mol. Spectrosc. 1962, 8, 315.
0022-3654/84/2088-2697$01.50/0
been carried out on the properties of excited rare-earth (RE) coordination compound^.^ UP to the present, most of these studies (3) Schimitschek, E. J.; Schwarz, E. G. K. Nature (London) 1962, 196, 832. (4) Lempicki, A,; Samelson, H. Phys. Lefr. 1963, 4, 133.
0 1984 American Chemical Society
