Ind. Eng. Chem. Fundam. 1983, 22, 116-121
116
resulted from incomplete conversion of carbon in the char. Sulfur dioxide levels reached nearly 2000 ppm in the reactor. Acknowledgment This research was supported initially by the US.Department of Energy, Washington, DC, with Dr. Robert Wellek as project officer. Subsequently, support was received from the U S . Department of Energy, Morgantown Energy Technology Center, with Dr. Sirgit Singh and Mr. Gary Friggens as project officers. The assistance of Jeffrey R. Burkinshaw, L. Douglas Skinner, and Wesley Pack in testing and analysis is recognized. Registry No. 02, 7782-44-7; HCN, 74-90-8; NH3, 7664-41-7; NO, 10102-43-9; H2S, 7783-06-4; S02, 7446-09-5; N,, 7727-37-9; S, 7704-34-9.
Literature Cited
Farnsworth, J. F.; Mitsak. D. M.; Kamody, J. F. EPA Meeting for Environmental Aspects of Fuel Conversion Technology, St. Louis, MO, May 1974. Hansen, L. D.; Phillips, L. R.: Mangelson, N. F.: Lee, M. L. Fuel 1980, 59, 323. Luthy, R. G. DOE Report No. FE-2496-16, DOE Contract No. EX-76-S-012496, Carnegie-Mellon University, Pittsburgh, PA, f978. Make, P. C.; Rees. D. P. I n "Pulverized Coal Combustion and Gasification", Smoot, L. D.; Pratt, D. P., Ed.; Plenum Co: New York, 1980; p 183. Massey, M. J.; Dunlap, R. W.; Koblin, A. H.; Luthy, R. G.: Nakles, D. V. DOE Report No. FE-2496-3, DOE Contract No. EX-764-01-2496, CarnegieMellon University, Pittsburgh, PA, 1878, Oldham, R. G.; Wetherold, R. G. DOE Report FE-1795-3. Part 3 DOE Contract No. EX-76-C-01-1975, Austin, TX, 1977. Price, T. D. M.S. Thesis, Chemical Engineering Department, Brigham Young University, Provo, UT, 1980. Skinner, F. D.; Smoot, L. D.; Hedman, P. 0. American Society of Mechanical Engineers, New York, 1980; ASME Paper 80-WANT-30. Skinner F. D. Ph.D. Dissertation, Department of Chemical Engineering, Brigham Young Unlversity, Provo, UT, 1980. Smoot, L. D.; Hedman, P. 0. EPRI Report No. FP-806, EPRI Contract No. 304-1. Brigham Young University, Provo, UT, 1979. Waish, P. M. DOE Report No. FE-2762-2, DOE Contract No. EF-77-S-012762 Princeton University, Princeton, NJ, 1978.
Burkinshaw, J. R.; Smoot, L. D.; Hedman, P. 0.; Blackham, A. U., submitted for publication in Ind. Eng. Chem. Fundam. Bissett, L. A. Report No. MERCIRI-7872, U S . DOE Morgantown Energy Research Center, Morgantown. WV, April 1978.
R e c e i v e d for reuielv June 15, 1981 R e v i s e d m a n u s c r i p t r e c e i v e d September 28, 1982 A c c e p t e d October 4, 1982
Law of Corresponding States of Uni-univalent Molten Salts Makoto Harada, Masataka Tanlgakl, and Yutaka Tada' Institute of Atomic Energy, Kyoto University, Uji, Kyoto, Japan
A method to yield a law of corresponding states for alkali halides was developed. The complicated pair potential
between the constituting ions was simplified to Coulomb and soft-core potentials. The state functions of real molten salts was mapped to the domain of the reference salt, whose ions Interact through the hard-sphere core potential and the Coulomb potential by scaling the soft-core potential. The validity of the mapping was demonstrated for potassium chloride by use of the computer simulation of Woodcock et ai. The corresponding-states correlation for the thermodynamic properties of alkali halides was obtained in the mapped space. A simplified corresponding-states correlation using the characteristic temperature and the molar volume was also discussed.
Introduction The approach of the corresponding-states correlation is important for predicting the properties of molten salts. Ions in the molten salts interact through many kinds of potentials. The predominant long-range contribution to the ion pair potential arises from the interaction due to charge, i.e., the Coulomb potential. At short distance, the repulsive force operates. Further, the weak long-range potentials, e.g., dipole and the quadrupole interactions, should be taken into account. Reiss et al. (1961)examined the corresponding states for molten salts on the basis of two assumptions. One is that ions interact through the Coulomb potential and a core repulsion potential, which is the same for unlike ions and for like ions. The other is that the core potential can be described by the hard-sphere potential. The first assumption by Reiss implies that a configuration in which short-range repulsion between like ions comes into play contributes little to the configuration integral. Woodcock et al. (1971) reported from the results of Monte Carlo computer experiments on molten potassium chloride that this assumption is reasonable. Tottori University, Tottori Japan.
Blander (1967) employed Reiss's first assumption and developed the treatment to the molten salts interacting through the pair potential of conformal type. He postulated that the core potentials in a class of uni-univalent molten salts such as alkali halides differ only by a scaling factor of ions, and the softness of the core potential is kept unchanged in that class of salts. The softness of the core potential and the weak longrange potentials are different for each molten salt and these are important for understanding the properties of molten salts in a wide range of state variables. The present objective is to propose a method which yields a law of corresponding states for uni-univalent molten salts interacting through the Coulomb and core potentials together with the weak long-range potentials. The configuration integral is a function of the softness parameter and the parameters defining the long-range potentials besides the state variables, which are the reduced temperature and the reduced density. If the state functions of molten salts can be mapped onto the state space of a reference molten salt with particular values of these parameters, the corresponding states would be obtained in the mapped space. We select a hypothetical molten salt interacting through the Coulomb and the hard core potentials as the reference salt. The weak long-range
0196-4313/83/1022-0116$01.50/00 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
potentials are taken into account as modifying the Coulomb attractive potential. For mapping, the effective diameter of ions is required. This effective diameter is determined by a perturbation method with the help of graph theory. It will be shown that the resulting corresponding-states theory successfully correlates the thermodynamic properties of molten alkali halides. It is also shown that a simpler law of corresponding states with two parameters, characteristic temperature and molar volume, gives satisfactory correlations for thermodynamic properties.
Scaling of Soft Core We postulate the mi-univalent ionic salts comprised of N / 2 cations and N / 2 anions. The configuration integral is given as Z=
i>j
The integration is over the volume V of the salt. k and T a r e the Boltzmann constant and the temperature, respectively. 4ijis the pair potential between ith and j t h ions. The pair potential may be described as
+ zizje2/rc- f ( r )
= Xije-'/P
(2)
The first term represents the repulsion core potential of the Born-Mayer type with the parameters Xu and p . The second term is the Coulomb potential, where zi and t are the electrical valence of ith ion and the dielectric constant, respectively. The third term represents the weak longrange attractive potential. This attractive potential is much weaker than the Coulomb potential, and the configuration of the ions is mainly determined by the first and the second terms. Thus, the pair potential of eq 2 can be rearranged to give eq 3 as an approximation.
4 i j ( r ) = Xije-'/P + zizje2E/r
(3)
Here, E is a parameter specific to the salt species, which involves the effects of the weak long-range potential and the dielectric constant. The excess free energy, the free energy less its ideal contribution, is designated as AB' for a molten salt. Aex/NkT I Aex(T,V/N,A,/N
Xij,p)
T
Id
= zizje2E/r;
kTd/e2&
T
>d
(6)
Q E V/Nd3
(74
(7b)
If we can find the hard-sphere diameter, d, so that the relation Aer
= A*ex
=
HtAt 7txtE H
+...
The f bond is defined by f = exp(-&) - 1; P = l / k T
(10)
and it can be divided into two parts +fb
The f* bond is the one for the reference salt f* = exp(-@$*) - 1
(11) (12)
The f b bond is nonzero only in a small range of the interionic separation. This was called the blip function by Andersen et al. (1971). Introduction of eq 11 into eq 9 yields -PA"" = -pA*eX +
+ [the sum of simple irreducible diagrams involving at least two f b bonds] (13) where the dotted bond represents the f bond. If the second square-blacketed terms of higher orders are neglected and the effective hard-sphere diameter d is chosen so that the sum of the diagrams involving only one f b bond vanishes, the excess free energy is expressed by eq 8. Equation 8 then gives a simple thermodynamic equation expressing the law of corresponding states. The radial distribution function for the ions labeled 1 and 2, the position of which are at r l and r2,is expanded by a series of diagrams (Morita and Hiroike, 1961) g(1.2) =
(14)
where = is the e bond, exp(-&). The effective hard-sphere diameter is determined so that the f i s t blacketed term in eq 13 vanishes. From eq 13 and 14, the d value is given implicitly as
(5b)
A*ex(p,Q,A,)
AB=_ A,/Nd2
[sum of all simple irreducible diagrams composed of two or more black density circles and f bonds] (9a)
(54
The excess free energy of this reference salt, A*ex,is expressed as A*ex/NkT =_
=
(4)
A, represents the interfacial area between any two phases which happen to be in coexistence. Let us select a reference salt, which is composed of ions of the same size interacting through the Coulomb and the hard-sphere potentials
4 * i j ( ~=)
erence salt, and the law of corresponding states is satisfied in this space. The excess free energy can be expressed in terms of the diagram (Morita and Hiroike, 1961) as
f=f*
exp(-C&j/kT) dr'
117
(8)
holds, the thermodynamic properties of real molten salts can be mapped onto the state variable space of the ref-
where The difference between 4 and #J*arises from the core potential only. When the core potentials of the like and the unlike ion pairs are the same in eq 15, the term exp( P & j ) f b i j is invariant irrespective of the pair species. In this case, eq 15 becomes
where
118
Ind. Eng. Chem. Fundam., Vol, 22, No. 1, 1983
Table I. Interionic Potential Parameters and ( and f
LiF LiCl LiBr LiI Na F NaCl NaBr NaI KF KCI KBr KI RbF RbCl RbBr RbI CsF
2.67 3.78 4.41 1.90 4.17 20.1 16.4 9.79 8.39 28.6 44.7 44.h 15.5 72.6 64.7 108 98.5
2.99 3.3% 3.53 4.30 3.30 3.17 3.40 3.86 3.38 3.37 3.35 3.35 3.28 3.16 3.35 3.37 2.82
1121 883 823 742 1268 1074 1020 933 1129 1049 1007 9 54 1048 988 955 913 976
a Effective diameter evaluated from eq 18. estimated by Kirshenbaum et a1 (1962).
2.14 2.63 2.79 3.11 2.46 2.87 3.03 3.29 2.77 3.17 3.29 3.51 2.90 3.27 3.42 3.61 2.98
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 3.01
(17)
d+-(c)is the core part of d+-. Since ++-(') is expressed as A+ exp(-r/p), the d/p value is numerically calculated from eq 17 and the resultant values are fitted by the function d/P = 0.4069
0.970 0.988 1.005 0.982 0.929 0.974 0.972 0.967 0.931 0.963 0.967 0.968 0.947 0.959 0.956 0.967 0 960
1.67 1.72 1.64 1.58 2.00 1.90 1.90 1.85 1.95 1.90 1.90 1.90 1.90 1.89 1.89 1.85 1.94
Interionic distance used by Reiss e t al (1961).
Corresponding States for Alkali Halides It is very difficult to determine strictly the d value from eq 15 or 16, because the radial distribution function cannot easily be predicted at arbitrary sets of the state variables at present and an approximate method is necessary. The configuration in which short-range repulsion between like ions comes into play gives minor contribution to the configuration integral,because the short-range ordering of ions arises from strong Coulomb attractive potential. Thus, eq 16, in which the core potentials are taken to be common to the like and the unlike ion pairs, is first discussed in determining the effective hard sphere. Equation 16 has a blip function which is effectively nonzero only in the vicinity of r = d. On the contrary, g,(r) is not so sensitive to r in this range and it may be assumed as an approximation that g&) is constant irrespective of r. This assumption yields dS = 3Jm{exp(@$,_'c)) d - 1Jr2dr
0.827 0.801 0.845 0.858 0.867 0,947 0.948 0.959 0.893 1.010 1.014 1.024 0.905 0.982 0.989 1.038 0.867
+ 0.9075 In ( A + . / k T ) + 6.042 X 10-7X+-/kT (18)
Strictly speaking, the d value is dependent upon the density because the radial distribution function is dependent on the density. This minor density dependency of d / p is not included in eq 18. The d values evaluated from eq 18 at the melting point are shown in Table I for alkali halides, together with the interionic separation distances used for the correlation of the melting temperatures of alkali halides (Reiss et al., 1961). Let us consider the relationship between eq 15, 16, and 18. Woodcock (1972) have calculated from the result of molecular dynamics simulation the radial distribution function for potassium chloride near the melting point with the help of Tosi-Fumi potential parameters (1964). The d value calculated from eq 16 using the above radial distribution function is 3.19 x lo-* cm which should be compared with the corresponding value, 3.17 X IO-* cm, which is calculated from eq 18. The difference between these two values is reasonably small. The d value calculated from eq 15 with the help of the above radial distribution function is 3.06 X cm. This value is a little qmaller
4140 3080 3020 3250 4270 3400 3200 3160 3460 3200 3170 2980 3280 3140 3130 2980 2915
5.20 5.11 5.11 5.64 5.75 5.31 5.25 5.45 5.25 5.15 5.33 5.30 5.25 5.39 5.54 5.41 5.33
Critical temperature
than those evaluated from eq 16 and 18. This difference arises from the difference in size of the anion and cation. The radial distribution functions for the like and the unlike ion pairs are difficult to evaluate theoretically. Thus, we introduce a fourth parameter, {, which incorporates the effects caused by the difference in size of the anion and the cation. Then, the effective interionic distance is expressed as d/P =
fl0.4069 + 0.9075 In ( X + _ / k T )+ 6.042 X 10-7A+_/kTJ (19) The { value is close to unity and is taken as a constant value specific to the salt species, though this { value is temperature dependent, strictly speaking. Next, we consider the validity of the mapping relationship. The pressure of molten salts considered here can be mapped to the pressure domain of the reference salt with the use of the equation p * V / N k T = p V / N k T = function of ? and 0 (20) Here, the asterisk again represents the reference salt. The internal energy, U , of the salts, which is the sum of the energies due to the configuration energy and the kinetic one, can be mapped to the internal energy of the reference salt, U* I:*/NkT = /L'/NkT + 3 ( p * V / N k T - 1/2)Q]/(l + Q) = function of ? and 0, Q = d In d / d In T (21) This equation can be derived by differentiating the free energy relationship, eq 8, with respect to the temperature and by taking into account the temperature dependency of the d value. Woodcock et al. (1971) have performed the Monte Carlo computer simulation for potassium chloride, the ions of which interacted through the pair potential, eq 2 with t = 1. The results of the simulation are available for checking the mapped relationships of the present approach. Figure 1 shows the comparison of the mapped values for potassium chloride with those for the reference salt. The mapped values of the simulated internal energy are shown as the closed keys in Figure la. The mapping is performed using eq 21 with the potential parameters of 'rosi-Fumi (1964) and also with the parameter values, { = 0.963 and 6 = 1.01. These { and E va-lues were determined so that the mapped values of U at T = 0.0230 agreed
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 119 d I-
l
d
?
3
1
2
5
3
L
5
Figure 1. Comparison of the mapped values for potassium chloride with those of the reference salt. The closed keys are the mapped values of the internal energy and compressibilityfador by Woodcock et al. (1971).The open keys are the mapped compressibility factors of the experimental p-V-T relation by Goldman et al. (1976).The solid curves are the values of the reference salt.
with the values of the reference salt. The diameter evaluated from eq 19 with this { value is 3.05 X lo-* cm at the melting point and this agrees well with the one calculated from eq 15. The solid curves represent the internal energy of the reference salt obtained by Larsen (1974). The mapped values at other temperatures also agree with the ones of the reference salt within 1% deviation. The compressibility factors mapped to the reference salt space using eq 20 with the parameter values { and [ are compared in Figure l b with those of the reference salt, which are given by Larsen (1974) and Harada et al. (1982). The closed and the open keys respectively represent the mapped values of the simulated results and the experimental p-V-T relationship obtained by Goldman et al. (1976). The mapped compressibility factors are also close to the values of the reference salt (the solid curves). Taking into account the experimental precision of the compressibility factors of the reference salt, the mapping relation, eq 20, is satisfactory. These results suggest that the corresponding-states correlations for the thermodynamic properties are successfully obtained for molten salts. The molar volume and the pressure along the saturation curve can be related to the following universal functions, if the density dependency of the d value is neglected.
Q = o(f')
(22)
p d 4 / [ e 2 rj = rj(n
(23)
V / N d 3=
In similar fashion, the surface tension u and the isothermal compressibility K along the saturation curve are related as u d 3 / [ e 2z b =
i?(f')
(24)
~ [ e ' / d3~2 =
2(n
(25)
In eq 22-25, the d value is evaluated by eq 19. V is calculated from the density at low pressure which is given by Janz et al. (1967, 1974). The vapor pressure data were taken from Janz's handbook (19641, Lange (1961), and the handbook edited by Japan Chemical Society (1966). The { and { values for alkali halides were determined so that the experimental V and fi values at about l.lTmagree with those of potassium chloride. The values of { and { are shown in Table I. The temperature dependencies of V and fi for different alkali halides agree
- 1
1
A G
Figure 2. Corresponding-states correlation for the molar volume of alkali halides at normal pressure.
30
00
$.
5;
Figure 3. Corresponding-states correlation for the vapor pressure of alkali halides. The keys are the same as in Figure 2.
Figure 4. Corresponding-states correlation for the surface tension. The keys are the same as in Figure 2.
I
1
E
rl
Figure 5. Corresponding-states correlation for the isothermal compressibility. The keys are the same as in Figure 2.
with each other as is shown in Figures 2 and 3. Figures 4 and 5 show the corresponding-states correlations for the surface tension and the isothermal compressibility along the saturation curve, which are evaluated
120
Ind. Eng. Chem. Fundam., Vol. 22, No. 1 , 1983
Table 11. Characteristic Temperature and Molar Volume vo7
salt LiF LiCl LiBr
LiI NaF NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr
CsF CSCl CsBr CSI
T,,K 1246 987 962 889 1168 1048 994 933 1084 1049 969 909 1044 963 911 876 919 897 928
cms/mol 14.60 28.83 35.15 44.43 20.74 37.09 43.18 54.05 29.71 48.91 54.89 66.19 35.52 52.98 72.81 39.65 59.85 67.11 82.02
Fm 0.900 0.895 0.856 0.835 1.090 1.025 1.026 1.000 1.042 1.000 1.039 1.049 1.004 1.026 1.002 1.114 1.000 1.012 0.963
I
Fo
I
I
I
,
CSCl t
3.32 3.12 3.14 3.66 3.66 3.24 3.22 3.39 3.19 3.05 3.27 3.28 3.14 3.26 3.27 3.33 3.30 3.39 3.26
0.8
1
I
09
1.0
I I,3
1
1.1
i 2
T/TO
Figure 6. Simplified corresponding-statescorrelation for the molar volume of alkali halides at normal pressure. The keys are the same as in Figure 2, except for CsC1, CsBr, and CsI. .-3
with the helps of eq 24 and 25 with Tosi-Fumi potential parameters and the and E values given in Table I. The surface tension data were taken from Janz (1967,1977) and the compressibility data were from Bockris et al. (1957). These two properties can be correlated by eq 24 and 25 within 10% deviation (the surface tension of RbF is extraordinarily large and was excluded in the figure) and this fact ensures the validity of the present approach. The reduced melting temperatures, T,, of alkali halides are shown in Table I. A similar relationship can be observed except from lithium halides, though the melting behavior cannot reasonably be treated in the present approach. The critical temperatures of alkali halides, which are estimated by Kirshenbaum et al. (19621, and the resultant T, values are also listed in Table I. A similar relationship is again observed for alkali halides. Simplified Corresponding-States Correlation It was shown in the preceding section that the state functions of mi-univalent molten salta can well be mapped to the space of the reference salt, whose ions are equal in size and interact through the Coulomb potential and the hard-sphere potential. Since the ionic diameter is a weak function of the temperature, a constant value, which is specific to each salt, can be assigned in the limited temperature range as an approximation. Thus, the four-parameter law of corresponding states described in the preceding section can be reduced to two parameters, the characteristic molar volume, Vo,and temperature, T,. In this case, the molar volume and the pressure along the saturation curve are described as v/v, = (26)
I
0 8
O E 1
.c
0 -
-
Figure 7. Simplified corresponding-statescorrelation for the vapor pressure of alkali halides. The keys are same as in Figures 2 and 6.
1,n 1.8
I
1
1
I 2
1.0
1 4
T.'Tg
Figure 8. Simplified corresponding-states correlation for the surface tension. The keys are the same as in Figures 2 and 6.
v v(m
pVo/RTo 13 =p(m
(27)
where the reduced temperature is defined by
T =T/To
1 n
(28)
The surface tension and the isothermal compressibility along the saturation curve are described by
O ( V ~ / N , ) ~ / ~=/5~=T z(T') ,
(29)
KRTo/Vo= Z = z(m
(30)
We select potassium chloride as the standard salt. The melting temperature and the correspondingmolar volume of this standard salt are taken as Toand Vo,respectively. The Toand Vovalues for the other salts are determined so that the vapor pressure and the molar volume of these
..
1 2
1 4
Figure 9. Simplified corresponding-states correlation for the isothermal compressibility. The keys are the same aa in Figures 2 and 6.
salta fit those for potassium chloride. The resultant values
are listed in Table 11. Figures 6 and 7 show the corresponding-states correlations for the molar volume and-the vapor pressure. The temperature dependencies of V and for alkali halides agree with each other to yield the following universal relationships
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Q = 1.142 - 0.5570T + 0.4224P log
= 4.945 - 11.905/T - 5.033 log
(31)
T
(32)
The correlations of the surface tension and the isothermal compressibility are shown in Figures 8 and 9. These are expressed by the following equations within 10% deviation. 5 = 3.52 - 1.55T (33) P = 0.22 - 0.37F + 0.25P
(34)
The reduced melting and critical temperatures, T,,, and
TC,are shown in Table 11. The T, values for alkali halides are about unity except for lithium salts. The average value is
Tm = 1.03
(35)
if excluding lithium salts. The pctakes similar values for all alkali halides, and the average is
pc = 3.29
(36)
Young and O'Connel (1971) reported the corresponding-states correlations for the molar volume and also for the transport properties with the help of two parameters, characteristic temperature and molar volume. Their parameter values, which were determined from the molar volume and the thermal expansivity, do not give satisfactory correlations to the thermodynamic properties, especially to the vapor pressure. Conclusion A method to yield a law of corresponding states for the thermodynamic properties of uni-univalent molten salts was developed. The effective diameter of ions was evaluated by scaling the soft-core potential. Using this diameter, the state functions of real molten salts were mapped to the domain of the reference salt which interacts through the hard-sphere and Coulomb potentials. This gives the corresponding-states correlations of the thermodynamic properties. The effective diameter is a weak function of the temperature and is approximated to be constant. This approximation gives a simple law of corresponding states, which involves two parameters, i.e., characteristic temperature and molar volume. Universal functional forms of the simplified correlations for the molar volume, the pressure, the isothermal compressibility, and the surface tension of alkali halides along saturation curve were shown. Acknowledgment The authors gratefully acknowledge the financial support from a grand in aid for fundamental scientific research, Ministry of Education, Science and Culture, Japan. Nomenclature de,= excess Helmholtz free energy Aex = reduced excess free energy 4, = interfacial area A, = reduced interfacial area d = effective hard-sphere diameter e = elementary charge f = f bond f b = blip bond
121
f(r) = weak attractive potential excluding the Coulomb potential gij = radial distribution function between i and j ions g,, = radial distribution function, average value k = Boltzmann constant N = number of ions N A = Avogadro number p = pressure 15 = reduced pressure defined by eq 27 fi = reduced pressure defined by eq 23 R = gas constant r = interionic distance T = temperature T = reduced temperature defined by eq 28 T = reduced temperature defined by eq 7 To = characteristic temperature U = internal energy = volume, molar volume I!= reduced molar volume defined by eq 26 V = reduced volume defined by eq 22 Vo = characteristic molar volume 2 = configuration integral zi = valence of ith ion Greek Letters @ = l/kT { = parameter defined by eq 19 K
= isothermal compressibility
X = parameter of the soft-core potential $. = parameter modifying the reduced temperature p = parameter of the soft core potential u = surface tension & ' = pair potential between i and j ions $6) = core part of the pair potential
Subscripts c = critical point m = melting point +,- = cation and anion Superscript * = reference salt
Literature Cited Andersen, H. C.; Weeks, J. D.; Chandler, D. Phys. Rev. 1971, A 4 , 1597. Blander, M. Adv. Chem. Phys. 1967, 1 7 , 83. Bockris, J. O'M.: Richards, N. E. Proc. R . SOC.London Ser. A 1957, 247, 44. Goldman, G.; Todheide, K. 2.Naturforsch. 1976, 31A, 656. Harada, M.; Tanigakl, M.; Yao, M.; Kinoshita, M. J . Chem. SOC.,Faraday Trans. 2 1982, 7 8 , In press. Janz, G. J. "Molten Salts Handbook": Academic Press: New York, 1967, pp 39. 63. 80. Janz; G.J:;-&rdner, 0 . L.; Krebs, U.; Tomkins, R . P. T. J . phys. Chem. Ref. Data 1974. 3 . 1. Janz, G. J.: Tomkins, R. P. T.; Allen, C. B.; Downey, J. R., Jr. J . Phys. Chem. Ref. Data 1977, 6 , 409. "Kagaku Binran", Chem. SOC. Japan, Maruzen Press: Tokyo, 1966; p 555. Kirshenbaum, A. D.; Cahill, J. A.; McGonicagal, P.J.; Grosse, A. V. J . Inorg. Nucl. Chem. 1962, 2 4 , 1287. Lange, N. A. "Handbook of Chemistry", 10th ed.; McGraw-Hill Inc.: New York, 1961; p 1424. Larsen, B. Chem. Phys. Left. 1974, 2 7 , 47. Morlta, T.; Hlrolke, K. Progr. Theor. Phys. 1961, 25, 537. Reiss, H. S.;Mayer, S. W.: Katz, J. L. J . Chem. Phys. 1961, 3 5 , 820. Tosi, M. P.; Fumi, F. G. J . Phys. Chem. Solids 1964, 2 7 , 45. Woodcock, L. V. Proc. R . SOC.London, Ser. A 1972, 328, 63. Woodcock, L. V.; Singer, K. Trans. Faraday SOC. 1971, 6 7 , 12. Young, E.: O'Conneil, J. P. Ind. Eng. Chem. Fundam. 1971, 10, 418.
Received f o r review July 9, 1981 Revised manuscript received October 1, 1982 Accepted October 21, 1982
