LIQUID-LIQUID PHASE SEPARATION I N ALKALIMETAL-AMMONIA SOLUTIONS
3797
Liquid-Liquid Phase Separation in Alkali Metal-Ammonia Solutions.
V.
A Model for Two-Component Systems, with Calculations
by Paul D. Schettler, Jr., Patricia White Doumaux, and Andrew Patterson, Jr. Sterling Chemistry Laboratory, Yale University, S e w Haven, Connecticut Accepted and Transmitted by The Faraday Society
06520
(March 29, 1967)
An ionic-lattice model with long-range forces is proposed for alkali metal-ammonia solutions to allow calculation of the chemical potentials of both components in both phases as functions of a rationalized concentration scale in which ammonia molecules are presumed strongly to solvate the metal. From the chemical potentials both phase diagrams and vapor pressures can be calculated. The effect of varying the solvation number g and the parameters TI, V2,and D is examined. Of these, the dielectric constant has the most profound influence.
In a series of papers1-* we have investigated some experimental aspects of the phenomenon of liquidliquid phase separation in solutions of alkali metals in liquid ammonia. The electrical and quasi-metallic properties of these solutions are such as to indicate phase separation in them is not, in a number of ways, comparable to that) in other systems, essentially nonelectrolytic in character, which have heretofore been studied and interpreted in some detail.5-9 Several proposals have been made toward an understanding of phase separation in metal-ammonia solution,10-12though in no case has an entire phase diagram been calculated. In the present paper we undertake such a calculation. The results offer insight into our experimental measurements and assistance in planning further studies.
Phase Separation Model We propose to use an expression of the following form to give the chemical potential of the solute alkali metal in a concentrated metal-ammonia solution
pq/RT = In c‘
+ A’+;
(1)
where c’ represents moles of solute per effective volume, and A ’ is an appropriate Madelung constant. Its theoretical basis is the assumption, particularly appropriate to concentrated metal-ammonia solutions, of a quasi-crystalline ordered structure or ionic lattice; the Aladelung constant is used to determine the electrical potential in place of the Debye-Huckel potential func-
tion, and a c‘” concentration dependence results from the assumption of a lattice structure. Robinson and Stokes13note that an expression of this type will adequately represent various thermodynamic data for concentrated aqueous solutions. The chemical potential is a well-defined quantity which can be related to activity values; a theoretical calculation of the Nadelung constant can usually be avoided by algebraic manipulation of experimental activity data, which are often readily available. Equation 1 can be put in a (1) P. D. Schettler and 8 . Patterson, J . Phys. Chem., 6 8 , 2865 (1964). (2) P. D. Schettler and A. Patterson, ibid., 6 8 , 2870 (1964). (3) P. 1%‘. Doumaux and A. Patterson, ibid., 71, 3535 (1967), paper 111. (4) P. W. Doumaux and A. Patterson, ibid., 71, 3540 (1967). paper IV. (5) J. Houben and W. Fisher, J . Prakt. Chem., 123, 89 (1929). (6) H. L. Friedman and H. Taube, J . Am. Chem. Soc.. 72, 3362 (1950). (7) H. L. Friedman, ibid., 74, 5 (1952). (8) H. L. Friedman, J . Phys. Chem., 66, 1595 (1962). (9) E. 0. Eisen and J. Jaffe, J . Chem. Eng. Data, 11, 480 (1966). (10) K. S. Pitzer, J . Am. Chem. Soc., 80, 5046 (1958). (11) ?Z. J. Sienko, “Solutions MQtal-Ammoniac, PropriQtQsPhysicoChemiques, Colloque Weyl,” 11. J. Sienko and G. Lepoutre, Ed., UT.A. Benjamin, New York, N.P., 1964, p p 29-33. (12) N. F. Mott, Phil. Mag., 6 (62), 287 (1961). (13) R. A. Robinson and R. H. Stokes, “Electrolytic Solutions,” 2nd ed, revised, Butterworth and Co., Ltd., London, 1959, pp 225-226.
Volume ‘71, iVztmber 12 n’oiember 1567
P. D. SCHETTLER, JR.,P. W. DOUMAUX, AND A. PATTERSON, JR.
3798
form more useful for the metal-ammonia case. As Blumberg and Das have suggested,I4 ammonia is strongly bound to sodium, so solvent molecules thus bound should be considered part of the solute, not the solvent. Expressed in mole fractions C’ =
Nz’(V
- Nzb)
ih“=-“-----
---0
)r, -50-
(2)
where b is the impenetrable volume of a molecule and hence Nzb is the excluded volume. Accordingly C’ =
Nz’/(NI’CIi’
+ Nz’Vz’
- Nz’V2’)
(3) ~
The primed quantities thus represent a rationalized concentration scale defined as
Ns‘
N1’
=
= (Wl
+ Ni - yNz) - yNz)/(Nz(l - Y) + N1) Nz/(Nz
(4) (5)
where y is the number of solvating ammonia molecules strongly bound to a metal atom. Combining these expressions and incorporating the dielectric constant, as Pitzerio has done, we obtain (p2 -
p20)/RT
=
+
In ( N ~ I N I )
The equation has not been arrived at by rigorous derivation from first principles and hence may be regarded as empirical. The use of the bulk solvent dielectric constant is here mope acceptable than, for example, in the Debye-Huckel approach since it is believed the metal strongly binds amrnonial4 leaving the bulk ammonia unoriented and well described by the ordinary dielectric constant. We have satisfied ourselves in some detail‘j that eq 6 expresses data for concentrated aqueous alkali-halide solutions satisfactorily. The principal concern is that the solvation number, y, should remain constant over the range of concentration t o be considered. If the equation is to be used to calculate phase-separation curves, its most important property is that it predict, for cert,ain combinations of the parameters, that p2 is not single-valued. In Figure 1 pz is plotted as a function of N2’/N1’at reduced temperature and dielectric constant. It is found that pz is not single-valued, but that a range of p2 values is obtained such that three different concentrations have the same chemical potential. Further to determine the composition of the phases if a miscibility gap is predicted, one must involve the second component through the equality p1 = pl‘. The chemical potential of the solvent can be calculated from the Gibbs-Duhem equation, which simply The Journal
of
Physical Chemistry
N;h;
Figure 1. Plot of the chemical potential of a solute as function of concentration, expressed as mole ratio, calculated from eq 6 a t reduced temperature and dielectric constant. The dashed line is drawn to emphasize the loop in the curve which defines three values of Ar2‘/N,’ a t which f12 has the same value. The areas A I and A l are mentioned in the text in connection with eq 7 .
represents the area plothed against, the pz axis in Figure 1. The equation reduces t,o
where the primed quantities refer to the concentrated phase, the unprimed to the dilute phase. The only values of N2’/N1‘ and N z / N 1 which can be chosen consistent with the restrictions of eq 7 are the two with dpz/d(Nz/N1) positive such that the areas A i and Az in Figure 1 are equal and that the area under pz - p20 between N2/N1and N2‘/N1‘is zero. It then becomes possible to calculate phase-separation curves, so long as the necessary parameters can be evaluated. For metal-ammonia solutions all the data of eq 6 are available, excepting a value of the Madelung constant. As earlier stated, one approach is to take experimental activity values and by manipulation of eq 6 to obtain y and A . In preference to this we have chosen t o use the consolute data for phase separation for each metal as the only experimental datum to fix the phase-separation curve, since the only source of activity data is limited measurements of the vapor pressures of metal-ammonia solutions. Reference to Figure 1 and consideration of eq 6 and 7 show that a t the consolute point both the first and second derivatives of pz with respect to N2/N2are zero, as shown in eq 8 and 9. (14) W.E.Blumberg and T. P. Das, J. Chem. Phys., 30, 251 (1959). (15) P. D. Schettler, Dissertation, Tale University, New Haven, Conn., 1964; Dissertation Abstr., 25, 4432 (1965). Order No. 651949,University Microfilms, Inc., Ann Arbor, Mich.
LIQUID-LIQUID PHASE
SEPARATION I N ALKALI
J'IETAL-AMMONIA
I
By using, for each alkali metal studied, the experimental values of the consolute temperature and concentrations, one can evaluate the solvation number, y, and the Madelung constant, A , from eq 8 and 9. These experimentally derived values can then be used to caland hence a complete liquid-liquid phase culate pz - rzo, diagram for a particular metal. This has the effect of pinning the phase diagram computed with eq 6 a t a single experimental point, the consolute point.
Calculations To facilitate this procedure, and to make it easier to examine the effect on the phase-separation calculation of changing the parameters in the model, the equations have been programmed for machine computation. l6 The data for the calculations were arrived a t through a careful reexamination of all existing data on phase separation of lithium, sodium, and potassium, including those of Kraus and Lucasse, l7 Loeffler,l8 Frappd, l9 Schettler and Patterson,' and Doumaux and P a t t e r ~ o n . ~Our best estimates of the consolute points for sodium are given in Table I; the data for sodium only are represented by the smooth curve E in Figure 2. The values of TI and were determined from density data on pure ammonia20 and on sodium-ammonia solutions.*l A plot of for both sodium and potassium is slightly concave downward in the concentration region of the phase diagram; in initial calculations (see below) the value of the maximum was used. All values used in the calculation on sodium are shown in Table 11. Since the results for the other metals are quite similar to thosefor sodium, only those for sodium will be discussed; lithium and potassium are fully covered in ref 16.
v2
v2
Table I: Consolute Temperature and Concentration-Two-Component Systems
Sodium
3799
SOLUTIONS
"
"
"
I
.
,
.
I
,
0 0.01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 Nz/,lr,.
Figure 2. Phase-separation curves for sodium-ammonia solutions. The curve labeled E (solid line) is a plot, drawn as a smooth curve through a large number of points not shown, of data from a number of investigators.'~4~17-'9 The experimental points, which define the curve quite unambiguously, are omitted to avoid cluttering the graph. Curves marked I (solid line, toward left), 11, I11 (- - IT' (- . - * ), and T' (- - -) represent various stages of the calculation using eq 6, 7 , 8 and 0. The data used a t each step of the calculation are discussed in the text and are listed in Tables I and 11. -)j
temperature has the correct value, but the temperatureconcentration curve is found to be displaced toward a lower value of N 2 / N 1 = 0.0311. Refer to curve I, Figure 2 . 11. Both eq 8 and 9 were utilized, a solvation number of 3.99 was calculated, and the computation of the phase diagram repeated. See curve 11. As a result of fixing the consolute point data the curve passes through this point correctly, but is, in comparison to curve I, expanded along the concentration axis, though the shape is like that of the experimental curve, E , with the same sharp rise on the dilute side and a gradual decrease on the concentrated side. Since A was the last of the parameters determined, it is sensitive to all other changes. 111. Next, a value of T2was chosen at the Consolute concentration rather than the maximum value mentioned above; refer to Table I1 for the values used. ~~
Investigator
Tomsolute
(NdA'1)eonsoiute
Kraus" LoefRer18 This work
-41.6 -41.6 -41.55
0.0433 0.0445 0.0413
Calculations were made as follows. I. The Madelung constant was calculated with eq 9, setting y = 6; these values were used to produce the phase diagram. Since Tconsolute was used, the consolute
(16) P. W. Doumaux, Dissertation, Yale Cniversity, New Haven, Conn., 1967. (17) C. A. Kraus and W. W. Lucasse, J . Am. Chem. Soc., 44, 1949 (1922). (18) D. E. Loeffler, Dissertation, Stamford University, Palo Alto, Calif., 1949. (19) G. Frapp6, Dissertation, Universit6 Catholique de Lille, Lille, France, 1958. (20) "Internationrtl Critical Tables," E. W.Washburn, Ed., McGrawHill Book Co. Inc., New York, N. Y., 1928, Vol. 111, p 23. (21) C. A. Kraus, E. S. Carney, and W.C. Johnson, J . Am. Chem. SOC.,49, 2206 (1927).
P. D. SCHETTLER, JR.,P. w.DOUMAUX, AND A.
3800
Table I1 : Values of Parameters Used in Calculation-Two-Component
Sodium
Trial
A
VI
I I1 I11 IF'
- 39,360 - 39,060 - 36,050
0.02461 0.02461 0.02461
- 36,050 - 36,050
V,(T) V,(T)
v a
Units: A , (caI/moIe)(l./mole)"a;
PATTERSON,
JR.
Systems"
v*
?4
D
0 06679 0.06679 0.06595 0.06595 0.06595
6.00 3 99 4.01 4.01 4.01
22.5 22.5 22.5 22.5
D(T)
VI, 72, I./moIe.
sodium -3S.OO'C
-
Tr. Curve V 'was calculated in the same way as curve IV, but the dielectric constant was varied as a function of temperature in an arbitrary way. Burow and LagowskiZ2 have determined the dielectric constant of ammonia at several temperatures and quote values reported earlier by others. If one plots these data and draws a "best" straight line through them, the function D = 22.5 0.0970(231.6 - 5") is obtained. This function was used to specify the variation of dielectric constant with temperature in the phase separation calculation. The result is curve V , which is so sharply contracted along the concentration axis as to fall well within the experimental curve. The asymmetry of the curve, mentioned earlier, is lost.
+
Discussion It is evident from Figure 2 that the model adopted, when used with the consolute point fixed to experimental data for alkali metal-ammonia solutions, produced parabolic phase separation curves with close similarities to the experimental curve. The parameters in eq ti affect the calculated results in varying degrees, D being appreciably less, and the - most significant, 2~ less so, VI least of all. Because the calculation is so sensitive to variation of the dielectric constant, it would be very easy to cause curve IV (which was computed with the optimal values of the other parameters available from experimental sources) to coincide exactly with the experimental phase separation curve, simply by adjusting the dielectric constant. We have not done this because it did not seem proper, at this point, to manipulate an adjustable parameter while undertaking to justify a newly proposed model. It must be recalled that the dielectric constant values measured experimentallyz2are scattered and their temperature dependence i.: not yet precisely specified. At the same time, however. the changes which mould be required in the dielectric constant are small, fall well within the experimental uncertainty
v2
L& nL
Figure 3. Plot of the vapor pressures of sodium-ammonia solutions a t -35.00' shown as ln(p/po) us. n1~/iV1. The experimental points are plotted in the curve to the left, and the calculated values derived from eq 6 are on the right. For a discussion of the lack of agreement between the two, refer to the text.
The effect is negligible, curves I1 and I11 being indistinguishable. They are not differentiated in Figure 3. IV. The same values as employed in curve I11 are used, but was permitted to vary as a function of temperature as specified in ref 20. The shape of the curve is retained, but there is a slight expansion along the concentration axis, curve IV. The Journal of Physical Chemistry
~
(22) D. F. Burow and J. J. Lsgowski, "The Solvated Electron," Advances in Chemistry Series, No. 50, .%merican Chemical Society, Washington, D. C., 1965.
LIQUID-LIQUID PHASE SEPARATION IN ALKALIMETAL-AMMONIA SOLUTIONS
of the quantity, and would be required to vary smoothly in just the way evidence from aqueous solut'ions suggest it does vary in the presence of different concentrations of solute. In aqueous solutions at 25", Hasted, Ritson, and Colliez3found that the dielectric constant of water fell as the concentration of electrolyte increased. If a similar effect of the magnitude of that found in water in the presence of 1-1 electrolytes existed in sodium-ammonia solution, the effect would be to shift curve V toward a better fit with the experimental curve, continuing to accept the variation of dielectric constant with temperature which we have arbitrarily used. In view of the present state of knowledge of the dielectric constant, it seemed to us unwise to do more than affirm its important bearing on the calculated results and to seek better data on it. At the same time, however, the prediction of the model is that D is the most interesting parameter to examine for its influence on phase separation. As noted earlier, constancy of the solvation number over a range of concentration must obtain if eq 6 is to be used. I n metal-ammonia solutions there is considerable evidence for strong and constant solvation even at concentrations approaching saturation. The values of y which result from the fitting of the consolute point data, yxa = 4.01, Y L ~= 3.61, and YK = 3.37, do not follow the usual view that the smallest ion is the most solvated. In these solutions the metal which is solvated includes, beside the metal ion, the monomer of
3801
Blumberg and DasI4 and other species as well. Marshall has found a similar contradiction in studies of vapor-pressure lowering, 2 4 though with an order just the reverse: E(,Li, ?Sa. Unfortunately, the lowering of fugacity depends on concentration in such a way that the data for the several metals cross each other several times at different concentrations; it is not clear how an order should be specified. Also, the solvation numbers we obtain are for different temperatures, which may have an important bearing on their order. Since 1.1~can be calculated, it is possible to calculate vapor-pressure curves.*6 Figure 3 is an example for sodium. As with the phase separation curves, these duplicate the trend of the experimental data but fall short of agreement in absolute magnitude; the values could, again, easily be arranged to agree by adjusting the dielectric constant. There would be interest in using vapor-pressure data, if of sufficient precision and detail, to obtain separate values of the parameters to be used in the phase-separation calculation.
Acknowledgment. This work was supported by the Kational Science Foundation. The computations were made possible by financial support, from the Yale University Computer Center. (23) J. B. Hasted, D. M. Ritson, and C. H . Collie, J . Chem. Phys., 16, 1 (1949). (24) P. 12. Marshall, J . Chem. Eng. Data, 7 , 399 (1962).
volzime 71 X'umber 12 XovembeT 1967
