DONALD A. MCQUARRIE
1508
lattice model can give qualitative agreement with experiment only if the complicated interplay between short range repulsive forces and long range coulomb forces is taken into account. Two alternatives remain for the calculation of the heats of more concentrated fused salt solutions. The rigid lattice model can be retained, and simplifying assumptions which mould allow a calculation of the coulomb energy can be sought, or the rigid lattice model can be abandoned, either by the introduction of a large concentration of vacancies or by the use of a radial distribution function theory. Either of these approaches is likely to be difficult.
IV. Conclusions The standard order-disorder theory was shown l o present some rather difficult conceptual prob-
Vol. 66
lems when applied to fused salt solutions primarily because interactions are limited to nearest neighbors, while the interactions in the ionic medium must extend to large distances. This does not invalidate the order-disorder approach, but does place some rather severe restrictions on it. Also, as \vas to be expected, the excess entropy and volume characteristics of the fused salt solutions which have been discussed are in almost total disagreement with the predictions of order-disorder theory. An attempt was made to calculate heats of fused salt solutions from existing knowledge of interionic interactions using a rigid lattice model. The theory was applied with semiquantitative success to the dilute solution, but could not be extended to more concentrated regions.
THEORY OF FUSED SALTS BY DONALD A. MCQUARRIE Department of Chemistry, University of Oregon, Eugene, Oregon’& Received March 17. 196.9
The LennardJones-Devonshire theory of liquids has been applied in a straightforward manner to systems of fused salts. Because of the long-range coulombic potential, the calculations must be extended to include all neighboring shells, but this problem is resohed by the natural occurrence of the Madelung constant. An interesting result is that the Coulomb potential does not influence the motion of an ion directly, and so a molten salt may be approximated as a fluid of hard spheres in certain cases. The derived partition function is used to calculate several thermodynamic quantities for fused salts. An equation of state is derived which obeys the law of corresponding states for fused Salt5 and from this the reduced critical constants of alkali halides are obtained. The treatment has been carried out only for the halides of lithium, sodium, potassium, and rubidium, but is easily extended to include other salts.
I In this article we shall apply the LciinardJones-Devonshire theory of liquids2 to fused salts. It will be shown that a straightforward application of their t,heory is possible in spite of the long-range coulomhi c potential . The most rigorous treatment of fused salts has been given by Stillinger, Kirkmood, and Wojto~ i c z .They ~ used the theory of distribution functions and derived the usual set of integral equations which, in their case, determines the fused salt ion distributions. It is interesting to note that the superpositioii approximation has not been used in t’heir m7ork, and a less restrictive hypothesis has been used in its place. Their treatment is quite formal. Carlsoii, Eyriiig, and nee5 have applied the rccent theory of significant structures to fused salts with success. In this theory, a partit,ion function is constructed by identifying three significant struc(1) Supported in part by a research grant from the National Science Foundation. (la) Department of Chemistry, Michigan State University, East Lansing, Michigan. (2) J . E. Lennard-Jones and A. F. Devonshire, Proc. Roy. Soc. (London), 8163, 53 (1937): 8166, 1 (1938). (3) F. H. Stillinper, J. G. Kirkwood, and P. J. Wojtowioz, J. Chem. Phya., 82, 1837 (1960). ( 4 ) For a discussion of the theory of niolec~hardistribution functions, see T. L. Hill, “Statistical Mechanics,“ McGraw-Hill Book Co., Inc., New York, N. Y., 1956, Chap. 6. ( 5 ) C. AI. Carlson, H. Eyring, and T. Ree, Proc. Natl. Acad. Sck., 46, 333 (1960).
tures in the liquid state. These are (1) solid-like degrees of freedom, (2) degrees of freedom arising from the possible position degeneracy of the solid, and (3) gas-like degrees of freedom. This is not rigorous but has a certain intuitive appeal. Recently an interesting criticism of this theory (and all other solid-like theories of the liquid state) has been published by Hildebrax~d.~Blomgren has extended the significant structure theory and has applied it to fused salts.’ The advantage to using the Lennard-JonesDevonshire theory is that it is fairly easy to produce numerical rcsults and that Kirkwood has given this theory a firm statistical mechanical foundation.8 This cell or cage model of liquids should be applicable to fused salts since i t requires a central molecule to be surrounded by its neighbors, “smeared” over the surface of a spherical shell whose radius is equal to the distance between the central molecule and its neighbors. In fused salts, the force between nearest neighbors is of the order of a hundred times stronger than that between rare gas molecules, to which the theory has been most extensively applied. It was found in reference 3 that a given ion is surrounded on the average by concentric sheIls of alternating charge, similar to that of the corresponding ionic crystal. This is exactly the model required by an applica(6) J. H. Hildebrand and G . Archer, ibid., 47, 1881 (1961). (7) G . E. Blomgi*en,Ann. N . Y. Acad. Sci., 79, 781 (1960). (8) J. G . Kirkwood, J . Chem. Phys., 18, 380 (1960).
TEEEORY OF FUSBD SALTS
August, 1962
tion 6f the Lennard-Jonewllevonshire theory. This l‘smeared on the average” concept is the same as that encountered in the familiar Debye-Huckel theory . The thleory is essentially a, theory of the solid state made applicable to the liquid state by introducing a disordered effect duc to thermal motion. Each molecule instead of vibrating around a lattice site is now allowed to move freely in the cell formed by its first neighbors. The entropy of an actual liquid would be higher than that calculated from this model, since each molecule is confined to its cell. Adclitional entropy can be introduced into this model by the inclusion of a suitable term in the partition function. This problem, called the communal entropy problem, is discussed elseThe only deviations that we shall make from the simple Lennard-Jones-Devonshire theory is t o consider st two component sysitem (the positive ions and negative ions), and an extension t o include shells of further neighbors, Buehler, Wentorf, Hirschfelder,. and Curtiss10 considered second and third shells 111 Q numerical tabulation of several important functions of the Lennard-Jones-Devonshire theory using a 6-12 potential. Thermodynamic properties are not greatly affected by the inclusion of higher shells when a 6-12 potential is used, but we must consider all shells, since the l/r coulombic potential decreases so slowly. I n section TI we shall derive an explicit expression for the canonical ensemble partition function, in section 111 we shall derive several thermodynamic quantities from this partition function and calculate entropies of fusion and vaporization and critical points for some 1-1 salts, and section IV will be a discussion of these results.
I1 For simplicity we shall consider only 1-1 type salts, but an extension to include others follows immediately. Since the Lennard-Jones-Devonshire theor,y is actually a modified theory of the solid state, we shall begin with a partition function, Q, for a two-component lattice and show how this may be modified to correspond t o the liquid state
g(N+, N-, N+-, N+ .--,.. )e -w/RT
1509
and negative ion8 are distributed as in the crystal. This is the point at which we cauld consider the existence of holes in fused salts. A ’ e shall not do this, however, but shall discuss this point later. This makes g equal to two and TY
+
3-
. .)
(3) where cj is the number of j-nearest neighbors and uj is the potential between j-neighbors. In order to apply this to liquids, we consider the particle not to be restricted to a lattice site but allowed to move in a volume bounded by its first neighbors. To allow for disorder due to the thermal agitation, the neighbors are considcrcd t o be uniformly distributed owr the surface of a sphere. We also include a factor for the communal entropy. The partition function now becomes = N+(ClUl
CZl.12
,
Q(N+,N-,T) =
where vf is the volume in which any ceiitral ion can move and h A = (5) (2nmlc T)‘le where h is Planck’s constant, m is the mass of a particular ion, k is Boltzmann’s constant, and T is the absolute temperature. The exponential term is still the energy of the system when all the ions are situated at lattice sites. The N+ ! and N- ! are the entropy factors. Note that this partition function is formally similar to that of an Einstein crystal, except for the factorials. We need only define 2rf to specify completely our partition function. vf is defined as
or for a spherically symmetric potential
(7) I n Fig. 1 the central ion is a t P, a distance r from the center of the shell formed by its j-th neighbors. The area of the .ring shown on the surface of the sphere is 2naj2 sin Ode. The number of neighbors in this area is
(1) where q I ( T ) is the partition function of a =!= ion situated a t a lattice site, N4 is the number of I ions, g is the number of ways N + positive ions and 2naj2 sin Ode Cj Cj = - sin sde (8) N-negative ions can be arranged on a lattice such 4raj2 2 that there are N+- positive and negative ions as first neighbors, N+.- as second neighbors, etc., and and the potential energy of interaction between the where ion a t P and the j-th neighbors in the ring is u(Rj)Cj sin Bdt3/2 where Rj2 = r2 ai2 - 2ajr cos 8. = N++u,.+ N--u, N+-u+. . . (2) The total interaction energy between the ion a t P u is the coulombic potential between two ions and all of its cj j-th neighbors in the shell is situated a t appropriate lattice sites. We assume that all the sites are occupied and that the positive cpi(r) -= 5 u(Rj) sin Ode (9)
w
+
+
+
+
J“
(9) T. L. Hill, “Statistical Thermodynamics,” Addison-Wesley Publishing Go., Ino., Reading, Mass., 1960. (10) R. H. Wentorf, R. J. Buehler, J. 0. Hizsohfelder, s a d C. F. Curtiss, J . Chem. Phye., 18, 1484 (1050).
2 0
The total interaction energy between the ion a t P and al1,of its shells is
1518
- + e 3
r
< (I .-
D
r
>a -
u
(13)
m h m a is the Madclung constant of the particular lattice, in the present case, simple cubic. This, of course, is the potential inpide a wiiformly charged sphere and could have becn mitten down ~ m mediately. It is this natural occurreece of the &ladelung constaiit which eliminates any pmhlem due to the coulombic potential. This simpIe result indicates that the Coulomb pstential, aside from scaling thermodynamic quantitiefi, does not ly influenre the motion of ttn ion in its cell. 1herefare, a molten salt may be approximated by a hard sphere fluid in certain cases. This has, iu fact, been done with some success iii the past, and the fact that the potential h i d e a uniformly charged sphere is constant furnishes an explanation for its success. The applicability of this f a d is made clear by the present treatment. ,7
vf now becomes
Fig. 1 .-A
I central ion and its j-t,h shell,
4 - 1 = ic PJ(4 =l M
(10)
We must now assume a form for a(&). We first assume that it consists of tn-o parts, a short-range potential and a long-range potential. As indicated above we assume that the long-range contribution is coiilombic and can be written as
where x is the valence of the ion, E is the charge on the electron. The short-range contribution should introduce a strong repulsive force at small values of r , indicating the improbability of the ions to int'er-penetrate. It would perhaps be most suitable t o assume a Lennard-Jones type of potential, hiit €or convenience we shall ignore any short-range attractive part and use a hard sphere potential for t hc repnlsiw part. u ( K J is then ?*.(R,) = (-1)'
=+,
~~c'/Rj
T
a-u
where v" = c3. This form for vf occurs in an approximation of the Lennard-Jones-Devonshire theory due to Prigogine and I\lathot.'l The occurrence of this form in the present development is not due to an approximation, but occurs because the electrostatic poteiitial inside a m i formly charged sphere is a constant. The partition function for the model is
From tthis we can derive all t,he thermodynamic properties of the model. Some will be derived in the next section. I11 We shall derive the equation of slate for this model. It is well kirown from statistical mechanics that'
(T
This immediately yields (12)
where sigma is the sum of the radii of the positive and negative ions. Let us assume that the crystalline form of the wystal state of the liquid is simple cubic. Most alkali halide salts have this form. p(r) then becomes
This equation of state consists of two parts, a hard sphere part and one due to the energy of the system when all the ions are situated on lattice sites. It, (11)
I. Piigogine and V. Mathot. J. Chrm, Phus.,
2 0 , 40 (1952).
August,, 1962
THEORY OF FUSED SALTS
has been pointed out by Bockris and RichardsI2 that fused salts obey an equation of state of hard spheres fairly well. This is an example of the successful approximation of a fused salt as a hard sphere fluid. This provides some evidence that our simple model may be satisfactory for fused salts. Equation 17 may be rearranged to illustrate tJhe law of corresponding states for fused salts as stated by Reiss, nlayer, and Katz:.la They have shown that the reduced variables for fuses saJts are $E~/GJCTand v*/v. The equation of state in these reduced variables is
20
1511
\‘
18
I (i
14
s 12 W.
X F-i 10
In Fig. 2 we have plotted this reduced equation of state. In it we have plotted pv*/kT vs. v/v* for T . this we were able to deterfixed ~ ~ e ~ / ( r l cFrom mine the critical constants for fused salts. The critical constants are
FL
8
6
4 2
which give for computation
0
(20) If u has units of Angstroms, !PC is in OK., V , in ~ m . and ~ , P, in atmospheres. The values of the critical constant)s calculated from these equations are listed in Table I. Carlson, Eyring, and Reej
Salt
LiF LiCl LiBr
LiI NaF NaCl 9aBr
NaI KF KC1 KBr KI RbF RbCl ItbHr ItbI
TABLEI C4LCVLATED CRITICAL CONSTANTS T~ (OK.) V~ (om.3) a (A.) 2 01 6350 122 2 57 4960 254 2 75 4640 312 3.02 4220 413 2 31 5530 185 2.81 4540 333 2 98 4280 398 3 .23 3950 505 2 87 4780 286 3 14 4060 464 3 29 3880 534 3 53 3610 660 2 82 4520 336 3 29 3880 534 3 43 3720 608 3 66 3490 735
p , (atm.)
910 343 261 180 525 240 190 137 294 154 128 96 236 128 108 83
also have cdculated critical constants and their results are compared with those of the present paper in Table 11. They report values for only four salts. The values of Carlson, Eyring, and Ree corresponding to eq. 19 are 16.7, 10.2 X and 13.4, respectively. These are average values for the four salts. Unfortunately these data are not (12) J. 0 A I Boclcris and pi. E. Richards f r o c . Rou. Soc (Londoni A241, 44 (1957) (13) H. Reiss S. W. hlager, and J. L. Kata, J . Chem. Phus., SS,
820 (1961).
,
++
0
I
I
I
I.
10
20 30 40 50 60 Reduced volume. Fig. Z.-pv*/lcT us. u/u*: curve 1, e2/aliT = 12.0; curve 2, e 2 / o k T = 13.1; curve 3, e2/akT = 13.5.
known experimentally. The agreement between our results and theirs seems to indicate that both calculated sets have some reliability. An advantage to the present approach is that there are no empirical parameters to fit to data. These are the only estimates of critical constants of fused salts of which the author is aware. TABLE I1 COXPARISON OF CRITICALCONSTANTS WITH CARLSON , EI~RING, AND REE’SRESULTS Tc
SaCl This paper C., E., and R. KCl This paper C., E., and R.
NaBr This paper C., E., and R. KBr This paper C., E., gnd R.
(OK.)
PO ( a h )
V , (ern
4540 3600
240 235 5
333 293
4060 3092
154 135 5
464 43 1
4280 3364
190 186 1
398 342
3880 3060
128 118 3
534 482
3)
Kow we shall derive the equation for the vapor pressure of a fused salt. We shall do this by equating the chemical potential of the liquid to that of the vapor in equilibrium with it. I t is known experimentally that alkali halides exist in the gas phase as diatomic molecules, so the vapor phase ill be considered to be an ideal diatomic gas,
1512
DOKALD ,4. MCQUARRIE
Miller and Kusch14 have found that vapors are associaid into clusters molecules, bn t we shall ignore this pjication. The chemical potential diatomic gas is
alkali halide of diatomic added coniof an ideal
Vol. 66
equal to the Madelung constant and t,he resuIts are listed in Table 111. As is to be expected, values of a somewhat less than this gire reasoilable agreement. TABLE I11 C A L C U L A T E D ENTIZOI’IES
O F TrAPORIZATION
Do
The chemical potential of the liquid phase is found from eq. 22
Setting pgas
= ,u+
+ p-
gives, after some algebra
(koaI./
a
salt
rnoIe)18
(B.117
LiF LiCl LiBr LiI ilaF PiaCl NaBr NaI
180.5 150 144 135 151 130 124 117.5 136 115 110 104 131 110 105.5 99.5
2.01 2.57 2.75 3.02 2.31 2.81 2.98 3.23 2.67 3.14 3.29 3.53 2.82 3.29 3.43 3.66
XF KC1 KBr
log P =
(M/WL+~-)” (8nii)1/2 ~ hi (7.6 uf2 e2
kT
x
KI
10-4)
+ 0.217 log T
RhF RhCl RhBr RbI
(23)
where P has units of mm., M is the mass of the diatomic molecule, I is its moment of inertia, v is its fundamental vibrational frequency, and Dois the energy necessary .to separate the moIecule into its constituent ions, not atoms, at OOK. Yote that this is not the usual definition of this quantity. Experimentally, vapor pressure curves for molten BIT C log T. salts are of the form log P = A This curve allows one to calculate several thermodynamic quantities from a knowledge of the molecular and crystal parameters. Equation 23 can be compared to experimentaI vapor pressure curves, but the values of vi are not known with sufficient accuracy t o be useful. Since the liquid does not possess the ordered structure of the solid, CY can hardly be expected to have the same numerical value as the Madelung constant of the crystal. Because of this, vf and a hai-e been treated as adjustable parameters and eq. 23 has been fitted to experimental curves. The data were taken from reference 15. For RCl, vf = 0.28 X loFz5cm.3 and CY = 1.1, and for XaBr, vf = 0 78 X 10-25 cm.3 and a = 1.1. Similar values would have been found for the other salts of this class. These values of uf difler from those found by Bockris and Richards” by a factor of approximately ten. Their experimental error was large however. Instead of varying cy and vf, we may investigate a alone by coilsidering the slope of log P vs. 1/T. If the boiling points are known, values of AS,, the entropy of vaporization, may be calculated and compared to experiment, This is a well-tabulated thermodynamic quantity and provides an indirect discussion of eq. 23. The value of CY has been set
+
+
(14) R. C. Miller and P. Kusch, J. Chem. Phus., 25, 860 f1956). (15) J 0%. Bockris, ed., “Modern Aspects of Eleotroohemis&r#y” Yol. 2. Beademic Preqs, Inc., New York, N. Y., 1959.
Tb (01
