1908
Ind. Eng. Chem. Res. 1997, 36, 1908-1920
Mixing Properties of Unsymmetrical Molten Salts Based on the Law of Corresponding States Yutaka Tada,*,† Setsuro Hiraoka, Yasushi Hibi, and Takao Kimura Department of Applied Chemistry, Nagoya Institute of Technology, Nagoya 466, Japan
The law of corresponding states of molten salt mixtures that include unsymmetrical salts was developed. The corresponding states equations for the thermodynamic and transport properties were obtained by expanding the Helmholtz free energy and the transport properties in terms of differences of the pair potential parameters, difference of the ionic size, and difference of the ionic mass. The effect of valences on the properties was accounted for by the products of cation and anion valences in the corresponding states equations. Equations for the mixing molar volume, surface tension, electrical conductivity, and viscosity were derived from the corresponding states equations. The estimated mixing properties satisfactorily agreed with the observed ones. 1. Introduction In the review of statistical mechanical theories of the thermodynamic properties of molten salts by Luks and Davis (1967), it was shown that Reiss et al. (1961) developed the theory of corresponding states for symmetric and unsymmetric pure molten salts with an ionic model through Pitzer’s dimensional analysis (Pitzer, 1939). In the ionic model, the pair potential is Coulombic interaction between like ions and is the sum of Coulombic interaction and hard-sphere repulsion between unlike ions. Polarization and dispersion effects are accounted for by an effective dielectric constant. The vapor pressure and surface tension for alkali halides were in the corresponding states. For the unsymmetric pure salts, the reduced properties were universal functions, but with respect to a particular value class. For binary molten salt mixtures with common anions, Reiss et al. (1962) introduced non-Coulombic interactions, polarization and dispersion, with coupling parameters to the pair potential. The excess free energy of mixtures was expanded around that of the reference salt, component salt 2, in terms of the difference between the coupling parameters and the difference between scaling parameters for length to give first-order and second-order approximations. Kleppa and Hersh (1961) and Hersh and Kleppa (1965) showed that the heat of mixing of alkali nitrate mixtures and alkali halide mixtures correlated well with the difference between the coupling parameters and the difference between the length parameters. Davis (1964) derived the excess free energy linear to the difference between the length parameters for unsymmetrical molten salt mixtures, alkali earth nitrate-alkali nitrate systems. The data of McCarty et al. (1964) on the alkali earth nitrate-alkali nitrate mixtures and the data of McCarty and Kleppa (1964) on lead chloride-alkali chloride mixtures and magnesium chloride-alkali chloride mixtures bore out the linear form. In those works, corresponding states equations for the transport properties of pure molten salts and equations for the mixing molar volume, surface tension, and transport properties were not developed. Young and O’Connell (1971) studied the corresponding states correlation of uniunivalent molten salts and their mixtures to get empirical and predictive correlations in terms of characteristic thermodynamic param†
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eters with good agreement of the calculated properties to the observed ones. However, corresponding states correlations of diunivalent molten salts and their mixtures and uniunivalent-diunivalent mixtures were not studied. Harada et al. (1983) simplified pair potential for molten alkali halides to the sum of soft-sphere repulsion and effective Coulomb potential which incorporated effects of polarization and dispersion. The soft-sphere potential was scaled to a hard-sphere potential such that the Helmholtz free energy of the molten salt with the simplified potential is equal to that of a hypothetical molten salt whose pair potential is the sum of the hardsphere potential and the effective Coulomb potential. The thermodynamic properties of pure molten alkali halides were correlated well in the corresponding states by using the hard-sphere diameter and the effective Coulomb potential parameter. Tada et al. (1988) showed that the transport properties of pure molten alkali halides correlated well in the corresponding states with the parameters of the simplified pair potential and the characteristic mass, which was obtained by expanding the transport property with the mass difference of the anion and cation. Tada et al. (1990a,b, 1992, 1993, 1995) applied the simplified pair potential to binary molten alkali halide mixtures and alkali nitrate mixtures. The corresponding states equations of the mixtures were derived from perturbation expansions of the Helmholtz free energy and the transport properties in terms of the difference between the ionic length parameters, that of the potential parameters, and that of the ionic masses. The mixing properties of the uniunivalent molten salts were estimated well by using the corresponding states equations. The aim of this work is to obtain the equations for the mixing properties of symmetrical (uniunivalent) and unsymmetrical (diunivalent) molten salts, alkali metal, alkali earth metal, amphoteric metal, and transition metal halides without using binary interaction parameters. The equations are useful even for the mixtures which do not have common anions or common cations. The simplified pair potential of Harada et al. (1983) is applied to the mixtures, and the corresponding states equations for the thermodynamic and transport properties are derived from the perturbation expansions of the Helmholtz energy and the transport properties. The effect of valences on the properties is accounted for by the product of the cation and anion valences. The © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1909
equations for the mixing properties are obtained from the corresponding states equations.
2
∑∑ l,m)1
ψ)(
2. Equations of Corresponding States
2
xlxmpClpAm) ∑∑ l,m)1
xlxmpClpAmψlm)/(
2
2.1. Thermodynamic Properties. A binary mixture of molten salt is considered:
(6a)
2
xlxmpλlpνmzλlzνmξlm)/(∑∑xlxmpλlpνm), ∑∑ l,m)1 l,m)1
zλzνξ ) (
λ, ν ) C, A (6b) zC1+
salt 1 ) CpC1
z
zA1-
ApA1
2
∑∑ l,m)1
1/F ) (
z
salt 2 ) CpC2 C2 ApA2 A2 +
-
It is postulated that the pair potential of ions i and j is expressed by eq 1, which was proposed by Harada et al. (1983)
φij ) ψij exp(-r/Fij) + ξijzizje2/r
(1)
where zi and e are the valence of ion i and elementary charge, respectively. ψij and Fij are parameters of core repulsive potential, and ξij is a parameter of effective Coulomb potential. A hypothetical soft-sphere molten salt reference and a hypothetical hard-sphere molten salt reference are considered, whose pair potentials are defined by eqs 2 and 3, respectively
φ°ij(r) ) ψ exp(-r/F) + zλzνξe2/r, λ, ν ) C, A (2)
2
xlxmpClpAm/Flm)/(
-A ˆ ) -βA ) -βA° ) -A ˆ°
(3)
where
∑∑ l,m)1
(7)
Assumption 2. The two or higher order terms of R and γ in the Helmholtz energy expansion contribute little to -A ˆ Rγ. Another pair potential with a perturbation parameter R is introduced.
φRij(r) ) ψij exp(-r/Fij) + [zλzνξ + R(zλizνjξij zλzνξ)]e2/r, λ, ν ) C, A (8)
) zλzνξe2/r; r > d, λ, ν ) C, A
2
(6c)
where ψlm, Flm, and ξlm are the parameters of the pair potential between the cation in salt l and the anion in salt m. Using eqs 6a-c and letting R ) γ ) 1 in the Helmholtz energy expansion with assumptions 1 and 2, which is stated below, makes the configuration Helmholtz energy of the mixture equal to that of the soft-sphere reference (see Appendix A).
φHij(r) ) +∞; r e d
zλzνξ ) (
xlxmpClpAm) ∑∑ l,m)1
2
xlxmpλlpνm), ∑∑ l,m)1
xlxmpλlpνmzλlzνmξlm)/(
λ, ν ) C, A (4) A pair potential with perturbation parameters R and γ is introduced
φRγij(r) ) [ψ + R (ψij - ψ)] exp{-(r/F)[1 + γ((F/Fij) - 1)]} + [zλzνξ R (zλizνjξij - zλzνξ)]e2/r, λ, ν ) C, A (5) where 0 e R e 1 and 0 e γ e 1. When R ) γ ) 0, the potential φRγij ) φ°ij reduces to that of the soft-sphere reference salt, whereas for R ) γ ) 1, the potential φRγij ) φij is that of the mixture of interest. The configurational Helmholtz energy or logarithm of configurational integral ln ZRγN of the system with the pair potential in eq 5 is expanded in powers of R and γ around that of the soft-sphere reference (eq A7 in Appendix A). The strong Coulombic interaction determines a locally ordered structure wherein cations are surrounded by anions and vice versa. Thus, we assume the following. Assumption 1. The interaction between like ions contributes little to the perturbed terms in the Helmholtz energy expansion. The potential parameters of the soft-sphere reference salt (characteristic potential parameters) ψ, F, and ξ are chosen to respectively be
The reduced configuration Helmholtz energy A ˆ can be expressed in terms of diagrams (Morita and Hiroike, 1961). The Mayer f bond in the diagrams can be divided into two parts, fH bond for the hard-sphere reference and fb bond which was called the blip function by Andersen et al. (1971). A ˆ R is double-expanded in powers of R and fb bond around that of the hard-sphere reference (see Appendix B). The characteristic separation distance d is chosen such that the perturbed diagrams that contain only one fb bond vanish. 2
∑∑ l,m)1
d3 ) (
2
xlxmpClpAm) ∑∑ l,m)1
xlxmpClpAmdlm3)/(
(9)
where dlm is derived from the configuration Helmholtz energy expansion of the pure salt and is given by (Harada et al., 1983)
dlm/Flm ) ζlm[0.4069 + 0.9075 ln(ψlm/kT) + 6.042 × 10-7(ψlm/kT)] (10) where ζlm is an ionic size difference parameter between the cation in salt l and the anion in salt m. Since fb bond is nonzero only in a small range of the interionic potential, the following is assumed. Assumption 3. The higher order terms of R more than 2 and those containing at least three fb bonds contribute little to the A ˆ R expansion. Using eqs 6b and 9 and letting R ) 1 in the A ˆR expansion with assumptions 1 and 3, the reduced Helmholtz energy of the mixture is expressed by eq 11 (see Appendix B).
ˆ A(T ˆ )Dd + G ˆ A(T ˆ )Dξ A ˆ ) βA ) A ˆH + C
(11)
1910 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
where
A ˆ H ) βAH
(12)
2
Dd )
xlxmxn[pClpAmpAn ((dlm3/d3) ∑∑∑ l,m,n)1 1)((dln3/d3) - 1) + pAlpCmpCn((dml3/d3) 1)((dnl3/d3) - 1)] (13) 2
Dξ )
xlxmpClpAm[zClzAmξlm/(zCzAξ) - 1]2 ∑∑ l,m)1
(14)
T ˆ ) kTd/(zCzAξe2)
(15)
Dd is the second-order perturbation term with respect to the difference of the core repulsive potential (ionic size difference), and Dξ is that with respect to the difference of the product of cation and anion valences and the effective Coulomb potential parameter. Dξ and Dd ) 0 for pure salts from the definitions 13 and 14. C ˆ A(T ˆ ) and G ˆ A(T ˆ ) are functions of the reduced temperature T ˆ. The pair potential between ions i and j of the hardsphere reference eq 3 is reduced with the characteristic parameter d and a characteristic parameter set zCzAξe2/ d.
φˆ Hij(rˆ ) ) φHij(r)/(zCzAξe2/d) ) +∞, rˆ e 1 ) zizj/(zCzArˆ ), rˆ > 1 (16) rˆ ) r/d
(17)
Equations 11 and 16 mean that the Helmholtz energy with the second-order perturbation terms Dd and Dξ follows the law of corresponding states. Thus, it can be said that the thermodynamic properties which are derived from the Helmholtz energy, i.e., molar volume, vapor pressure, and surface tension, follow the law of corresponding states when d and zCzAξe2/d are used as the characteristic parameters. These thermodynamic properties along the saturation curve can be related to the universal functions if the density dependency of the d value is neglected. 2
V ˆ ) V/[NA0
xl(pCl + pAl)d3] ) V ˆ H(T ˆ) + C ˆ V(T ˆ )Dd + ∑ l)1 G ˆ V(T ˆ )Dξ (18)
pˆ ) pd4/(zCzA ξe2) ) pˆ H(T ˆ) + C ˆ p(T ˆ )Dd + G ˆ p(T ˆ )Dξ (19) σˆ ) σd3/(zCzAξe2) ) σˆ H(T ˆ) + C ˆ σ(T ˆ )Dd + G ˆ σ(T ˆ )Dξ (20) where V ˆ H, pˆ H, and σˆ H are the reduced molar volume, vapor pressure, and surface tension for the hard-sphere moltent salt reference, respectively. The potential parameters used in the corresponding states equations are ψij, Fij, ξij, and ζij. For uniunivalent molten salts, alkali halides, the values of ψij and Fij were taken from the report of Tosi and Fumi (1964), and those of ξij and ζij were given in the report of Harada et al. (1983). The values of ψij, Fij, ξij, and ζij of diunivalent molten salts were determined by the least-squares fitting such that the observed values of the molar volume, vapor pressure, and surface tension of the pure salts obey the corresponding states equations V ˆ H(T ˆ ),
Figure 1. Corresponding states correlation for molar volume of diunivalent molten salts. Table 1. Interionic Potential Parameters and ξij and ζij salts
ψ, 10-10 erg
MgF2 MgCl2 CaF2 CaCl2 CaBr2 CaI2 SrF2 SrCl2 SrBr2 SrI2 BaF2 BaCl2 BaBr2 ZnCl2 CdCl2 CdBr2 CdI2 PbCl2 SnCl2
382 0.948 11.0 3.30 0.945 0.235 401 92.1 23.3 4.12 370 8.74 17.2 0.376 1.08 0.961 0.538 2.72 1.16
F, 10-9 cm 1.97 6.11 2.88 3.65 4.81 7.17 2.20 2.61 3.23 4.26 2.30 3.46 3.32 4.63 4.15 4.61 6.79 3.73 4.07
ξij
ζij
0.682 0.444 0.669 0.638 0.521 0.517 0.814 0.666 0.674 0.636 0.772 0.735 0.763 0.269 0.321 0.302 0.293 0.381 0.300
1.000 0.724 1.000 1.079 0.979 0.890 1.000 1.090 1.064 1.032 1.000 1.091 1.107 1.000 0.977 0.935 0.737 1.014 1.016
ˆ ), and σˆ H(T ˆ ), which were determined from the pˆ H(T corresponding states correlations of pure alkali halides. For pure salts, V ˆ (T ˆ) ) V ˆ H(T ˆ ), pˆ (T ˆ ) ) pˆ H(T ˆ ), and σˆ (T ˆ) ) σˆ H(T ˆ ) because Dd ) Dξ ) 0. In the determination of the potential parameter values, the observed values were weighted with factors of 3, 2, and 1 for the molar volume, vapor pressure, and surface tension from the point of the experimental errors. The molar volume of 19 diunivalent salts, the vapor pressure of 12 diunivalent salts, and the surface tension of 12 diunivalent salts were examined. The observed data are the recommended values of Janz (1967) and Janz et al. (1974, 1975, 1977, 1979). The determined values of ψij, Fij, ξij, and ζij are shown in Table 1. The equations V ˆ H(T ˆ ), pˆ H(T ˆ ), and σˆ H(T ˆ ) are given by eqs D1, D2, and D3, respectively, in Table 2. Figures 1-3 show corresponding states correlations of the molar volume, vapor pressure, and surface tension of the pure diunivalent molten salts. The root mean square deviations of the corresponding states correlations for the molar volume, vapor pressure, and surface tension were 0.22%, 5.76% (in log pˆ ), and 9.83%, respectively. MgCl2 in the surface tension and SnCl2 and PbCl2 in the vapor pressure and in the surface tension are not correlated well. The reason of the MgCl2 deviation may be the large polarizability of Cl- due to
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1911
Figure 3. Corresponding states correlation for surface tension of diunivalent molten salts. The keys are the same as in Figure 1.
∫0∞〈A˙ (0) A˙ (t)〉 dt
K ) β/V Figure 2. Corresponding states correlation for vapor pressure of diunivalent molten salts. The keys are the same as in Figure 1.
Table 2. Reference Terms and Perturbation Terms in the Corresponding States Equations Reference Terms ) 0.384 + 55.4T pˆ H(T ˆ ) ) 4.67 exp(-0.419/T ˆ) σˆ H(T ˆ ) ) 0.0289 exp(-46.1/T ˆ) κˆ *0(T ˆ ) ) 0.208 exp(-0.0339/T ˆ) ηˆ *0(T ˆ ) ) 0.0608 exp(0.0683/T ˆ) H(T ˆ)
V ˆ
(D1) (D2) (D3) (D4) (D5)
(21)
where A(t) is the dynamical quantity and A˙ (t) is its time derivative. The brackets 〈 〉 represent the canonical average. The transport property K is reduced with a characteristic mass and the characteristic parameters in eqs 6a-c and is expanded around that of the soft-sphere reference salt with the mass difference of the component ions, the difference of the potential parameters, the ionic size difference, and the difference of the product of cation and anion valences and the effective Coulomb potential parameter (see Appendix C).
Perturbation Terms
˜ Λ)1/2] K ˆ * ) Kd2Λ2/[A2(m
Uniunivalent Salts C ˆ V(T ˆ ) ) 0.336 - 5.75T ˆ C ˆ σ(T ˆ ) ) 0.0364 - 1.74T ˆ S ˆ *e2(T ˆ ) ) -6.60 × 10-2 + 1.79 × 10-3/T ˆ S ˆ *e4(T ˆ ) ) 2.72 × 10-2 - 1.68 × 10-3/T ˆ -4 Q ˆ *e1(T ˆ ) ) -0.298 + 6.93 × 10 /T ˆ Q ˆ *e2(T ˆ ) ) -8.06 + 0.143/T ˆ R ˆ *e(T ˆ ) ) -0.0669 + 2.64 × 10-4/T ˆ S ˆ *v2(T ˆ ) ) -13.1 + 0.366/T ˆ Q ˆ *v1(T ˆ ) ) -183 + 4.18/T ˆ Q ˆ *v2(T ˆ ) ) -2.57 × 104 + 412/T ˆ R ˆ *v(T ˆ ) ) 18.7 - 0.430/T ˆ
(D6) (D7) (D8) (D9) (D10) (D11) (D12) (D13) (D14) (D15) (D16)
Diunivalent Salts C ˆ V(T ˆ ) ) -2.93 + 165T ˆ G ˆ V(T ˆ ) ) 5.93 - 330T ˆ C ˆ σ(T ˆ ) ) 3.04 × 10-3 - 1.95T ˆ G ˆ σ(T ˆ ) ) -0.0803 + 3.02T ˆ fe(zCzAξ) ) -0.465 + 1.039 zCzAξ S ˆ *e2(T ˆ ) ) 0.246 - 4.56 × 10-3/T ˆ S ˆ *e4(T ˆ ) ) -1.47 + 0.0274/T ˆ R ˆ *e(T ˆ ) ) -1.11 + 0.0193/T ˆ G ˆ *e(T ˆ ) ) 1.29 + 0.0165/T ˆ fv(zCzAξ) ) -0.195 + 0.435 zCzAξ S ˆ *v2(T ˆ ) ) -13.7 + 0.277/T ˆ R ˆ *v(T ˆ ) ) -15.0 + 0.811/T ˆ G ˆ *v(T ˆ ) ) 76.2 - 2.36/T ˆ
(D17) (D18) (D19) (D20) (D21) (D22) (D23) (D24) (D25) (D26) (D27) (D28) (D29)
Uniunivalent-Diunivalent Salt Mixtures C ˆ V(T ˆ ) ) -0.0136 - 9.90T ˆ G ˆ V(T ˆ ) ) 0.00519 + 17.3T ˆ C ˆ σ(T ˆ ) ) 0.0693 - 3.29T ˆ G ˆ σ(T ˆ ) ) -0.0847 + 3.54T ˆ R ˆ *e(T ˆ ) ) -0.328 + 7.41 × 10-3/T ˆ G ˆ *e(T ˆ ) ) 0.694 - 0.0145/T ˆ R ˆ *v(T ˆ ) ) - 415 + 8.16/T ˆ G ˆ *v(T ˆ ) ) - 129 + 2.05/T ˆ
(D30) (D31) (D32) (D33) (D34) (D35) (D36) (D37)
the small ionic radius of Mg2+. The deviation of SnCl2 and PbCl2 will be discussed in the next section. The corresponding states correlations and the values of the potential parameters for alkali halides are shown in the report of Harada et al. (1983) and are not shown here. 2.2. Transport Properties. Collective transport properties, i.e., electrical conductivity, viscosity, and thermal conductivity, can be expressed with the help of the fluctuation-dissipation theorem
)K ˆ *0(T ˆ) + S ˆ *2(T ˆ )δ2 + S ˆ *4(T ˆ )δ4 + Q ˆ *1(T ˆ )ωF + Q ˆ *2(T ˆ )ωF2 + R ˆ *(T ˆ )Dd + G ˆ *(T ˆ )Dξ (22) where
Λ ) ψ exp(-d/F)/f(zCzAξ)
(23)
m ˜ is the characteristic mass of the mixture defined by 2
xl(pClmCl-1/2 + pAlmAl-1/2))/ ∑ l)1
m ˜ -1/2 ) (
2
(
xl(pCl + pAl)) ∑ l)1
(24)
where mCl and mAl represent the masses of the cation and the anion in salt l, respectively. f is a corrective function specific to the transport properties which accounts for the effects of the difference of the valences and the difference of the stoichiometric numbers of cation and anion in diunivalent salts. f is unity for uniunivalent salts and is a simple function of zCzAξ for diunivalent salt and the mixtures including diunivalent salts. The functional form of f is determined below. K ˆ *0 in the right-hand side (rhs) of eq 22 is the reduced transport property for the reference salt, whose ions have the unique mass m ˜ and interact through the reference potential φ°ij. The corresponding states equation for diunivalent salt mixtures has the same reference term K ˆ *0 as uniunivalent salts. The second and the third terms in the rhs of eq 22 are the second-order and the fourth-order perturbation terms with respect to the ionic mass difference, respectively. δ is the ionic mass difference parameter defined by
δ ) δC ) -δA
(25)
1912 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 2
δν )
xlpνlµνl, ∑ l)1
ν ) C and A
˜ /mνl)1/2 - 1 µνl ) (m
(26) (27)
where mνl is the mass of ion ν in salt l. The fourth and the fifth terms in the rhs of eq 22 are the first- and the second-order perturbation terms with respect to the mass difference and the core potential parameter difference. ωF is defined by eq 28 (Tada et al., 1990a,b).
Figure 4. Corresponding states correlation for electrical conductivity of diunivalent molten salts. The keys are the same as in Figure 1.
2
ωF )
xlxmpClpAm(F/(Flm - 1))(µCl + µAm) ∑∑ l,m)1
(28)
δ ) pCµC ) -pAµA and ωF ) 0 for pure salts. The last two terms in the rhs of eq 22 are the secondorder perturbation terms with respect to the ionic size difference and with respect to the difference of the product of cation and anion valences and the effective Coulomb potential parameter. Generally, K ˆ * and K ˆ *0 are given by functions of the reduced state variables, T ˆ and V ˆ . The reduced molar volume V ˆ along the saturation curve is expressed by the function of T ˆ , eq 18. Thus, K ˆ * and K ˆ *0 in eq 22 depend on T ˆ along the saturation curve. Application of eq 22 to reduced electrical conductivity and viscosity yields the corresponding states form of eqs 29 and 30.
κˆ * ) κd2(Λm ˜ )1/2/e2 ) κˆ *0(T ˆ) + S ˆ *e2(T ˆ )δ2 + S ˆ *e4(T ˆ )δ4 + Q ˆ *e1(T ˆ )ωF + Q ˆ *e2(T ˆ )ωF2 + R ˆ *e(T ˆ )Dd + G ˆ *e(T ˆ )Dξ (29) ηˆ * ) ηd2/(Λm ˜ )1/2 ) ηˆ *0(T ˆ) + S ˆ *v2(T ˆ )δ2 + Q ˆ *v1(T ˆ )ωF + Q ˆ *v2(T ˆ )ωF2 + ˆ )Dd + G ˆ *v(T ˆ )Dξ (30) R ˆ *v(T As the valences of anion and cation in the uniunivalent molten salts, alkali halides, are symmetrical, the odd order perturbation terms with respect to the mass difference δ2n-1 vanish from the definition of the characteristic mass m ˜ , eq 24 (Tada et al., 1988, 1990b). For pure alkali halides δ ) µC and ωF ) Dd ) Dξ ) 0. The plots of the reduced electrical conductivity against µC2 of the pure salts gave parabolic curves at any reduced temperatures, and those of the reduced viscosity gave straight lines. Thus, eqs 29 and 30 do not have terms higher than fourth-order and second-order, respectively. Functions of the reference terms κˆ *0(T ˆ ) and ηˆ *0(T ˆ ) and the perturbed terms S ˆ *e2(T ˆ ), S ˆ *e4(T ˆ ), and S ˆ *v2(T ˆ ) for alkali halides were determined from the observed data of electrical conductivity and viscosity of the pure salts by the least-squares fitting (Tada et al., 1988). The values of zCzAξ of the uniunivalent salts are almost unity. Thus, the corrective function f in the characteristic energy Λ, eq 23, is unity for the uniunivalent salts, as stated above. For the diunivalent molten salts, the reference terms κˆ *0(T ˆ ) and ηˆ *0(T ˆ ) are assumed to be the same as those for the uniunivalent molten salts so that all kinds of molten salts would be correlated in the corresponding states. The reduced electrical conductivity κˆ * and viscosity ηˆ * without using f decreased and increased linearly with increasing zCzAξ, respectively. Thus, the
Figure 5. Corresponding states correlation for viscosity of diunivalent molten salts. The keys are the same as in Figure 1.
functions fe for electrical conductivity and fv for viscosity were determined to be linear, and the coefficients of them were determined such that κˆ * and ηˆ * get close to κˆ *0 and ηˆ *0 by the least-squares fitting. As shown in Appendix C the reduced electrical conductivity and viscosity for the diunivalent molten salts were also expressed by eqs 29 and 30 as in the case of the uniunivalent molten salts for simplicity. The coefficients of S ˆ *e2(T ˆ ), S ˆ *e4(T ˆ ), and S ˆ *v2(T ˆ ) were determined by the least-squares fitting of the pure salts. ˆ ), ηˆ *0(T ˆ ), S ˆ *e2(T ˆ ), S ˆ *e4(T ˆ ), S ˆ *v2(T ˆ ), S ˆ *v2(T ˆ ), fe(zCκˆ *0(T zAξ), and fv(zCzAξ) are shown in Table 2. κˆ *0(T ˆ ) and ηˆ *0(T ˆ ) are given by eqs D4 and D5, respectively. S ˆ *e2(T ˆ ), S ˆ *e4(T ˆ ), and S ˆ *v2(T ˆ ) for uniunivalent salts are eqs D8, D9, and D 13. fe(zCzAξ), S ˆ *e2(T ˆ ), S ˆ *e4(T ˆ ), fv(zCzAξ), and S ˆ *v2(T ˆ ) for diunivalent salts are eqs D21-D23, D26, and D27. The electrical conductivity of 15 diunivalent salts and the viscosity of 7 diunivalent salts were examined. The observed data are the recommended values of Janz et al. (1974, 1975, 1977, 1979). Figures 4 and 5 show corresponding states correlations of the electrical conductivity and viscosity of the pure diunivalent molten salts. The root mean square deviations of the corresponding states correlations for the electrical conductivity and viscosity were 4.77% and 26.4%, respectively. CdI2 in the electrical conductivity and MgCl2 and BaCl2 in the viscosity are not correlated well. The reason of the deviation of MgCl2 may be the large polarizability of Cl- due to the small ionic radius of Mg2+. CdI2 is correlated well and SnCl2 and PbCl2 are not well in the thermodynamic properties, while SnCl2 and PbCl2 are correlated well and CdI2 is not well in the transport properties. Although CdI2, SnCl2, and PbCl2 have covalent effects, their opposite behaviors cannot be explained with this effect. The deviations might be due to the error in the observed data, but it is not clear at the present time. 3. Mixing Properties A mixing property ∆X for a binary system is defined by
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1913 2
∆X ) XM -
xlXl ∑ l)1
(31)
where XM is a property of a molten salt mixture and Xl is that of the component pure salt l. The mixing molar volume, surface tension, electrical conductivity, and viscosity are expressed by eqs 32-35 with the help of eqs 18, 20, 29, and 30, respectively.
∆Vcalc ) Nd3[V ˆ H(T ˆ) + C ˆ V(T ˆ )Dd + G ˆ V(T ˆ )Dξ] 2
NA0
xl(pCl + pAl)dll3V ˆ H(T ˆ l) ∑ l)1
(32)
∆σcalc ) (zCzAξe2/d3)[σˆ H(T ˆ) + C ˆ σ(T ˆ )Dd + G ˆ σ(T ˆ )Dξ] 2
xl(zClzAlξlle2/dll3)σˆ H(T ˆ l) ∑ l)1
(33)
˜ )1/2]}[κˆ *0(T ˆ) + S ˆ *e2(T ˆ )δ2 + ∆κcalc ) {e2/[d2(Λm S ˆ *e4(T ˆ )δ4 + Q ˆ *e1(T ˆ )ωF + Q ˆ *e2(T ˆ )ωF2 + R ˆ *e(T ˆ )Dd +
Figure 6. Comparison of mixing molar volume ∆Vcal calculated with Dd and Dξ to the observed ones ∆Vobs at x = 0.5 for (a) diunivalent mixtures and (b) uniunivalent-diunivalent mixtures.
2
ˆ )Dξ] G ˆ *e(T
xl {e2/[dll2(Λllm ˜ l)1/2]}[κˆ *0(T ˆ l) + ∑ l)1
ˆ l)(pClµCl)2 + S ˆ *e4,l(T ˆ l)(pClµCl)4] (34) S ˆ *e2,l(T ˜ )1/2/d2][ηˆ *0(T ˆ) + S ˆ *v2,l(T ˆ )δ2 + ∆ηcalc ) [(Λm Q ˆ *v1(T ˆ )ωF + Q ˆ *v2(T ˆ )ωF2 + R ˆ *v(T ˆ )Dd + G ˆ *v(T ˆ )Dξ] 2
xl[(Λllm ˜ l)1/2/dll2][ηˆ *0(T ˆ l) + S ˆ *v2,l(T ˆ l)(pClµCl)2] ∑ l)1
(35)
where
m ˜ l-1/2 ) (pClmCl-1/2 + pAlmAl-1/2)/(pCl + pAl) T ˆ l ) kTdll/(zClzAlξlle2)
(36) (37)
The functional form and the coefficients in the perturbation terms C ˆ V(T ˆ ), G ˆ V(T ˆ ), C ˆ σ(T ˆ ), G ˆ σ(T ˆ ), Q ˆ *e1(T ˆ ), Q ˆ *e2(T ˆ ), R ˆ *e(T ˆ ), G ˆ *e(T ˆ ), Q ˆ *v1(T ˆ ), Q ˆ *v2(T ˆ ), R ˆ *v(T ˆ ), and G ˆ *v(T ˆ ) were determined by the least-squares fitting of the mixing property data. The mixing molar volume of 9 diunivalent salt mixtures and 22 uniunivalentdiunivalent salt mixtures, the mixing surface tension of 3 diunivalent mixtures and 22 uniunivalent-diunivalent mixtures, the mixing electrical conductivity of 6 diunivalent mixtures and 9 uniunivalent-diunivalent mixtures, and the mixing viscosity of 2 diunivalent mixtures and 4 uniunivalent-diunivalent mixtures were examined. The observed data for the mixing properties are from Janz et al. (1974, 1975, 1977, 1979). As the uniunivalent mixtures are symmetrical, the valence effect terms G ˆ V, G ˆ σ, G ˆ *e, and G ˆ *v were not taken into account. For the diunivalent mixtures and uniunivalent-diunivalent mixtures, which are unsymmetrical, the valence effects terms G ˆ *e and G ˆ *v contributed to the transport properties more than the terms with respect to ωF, Q ˆ *e, and Q ˆ *v. Thus, G ˆ *e and G ˆ *v were taken into account, and Q ˆ *e and Q ˆ *v were set to be zero for simplicity. For the uniunivalent-diunivalent mixtures, the perturbation terms with respect to the mass difference S ˆ *e2(T ˆ ), S ˆ *e4(T ˆ ), and S ˆ *v2(T ˆ ) are given by the linear mole fraction combination of the corresponding terms of the component pure salts from
Figure 7. Comparison of mixing surface tension ∆σcal calculated with Dd and Dξ to the observed ones ∆σobs at x = 0.5 for (a) diunivalent mixtures and (b) uniunivalent-diunivalent mixtures. The keys are the same as in Figure 6.
the requirement that the mixing properties vanish for pure salts.
S ˆ *n(T ˆ ) ) x1S ˆ *n,1(T ˆ ) + x2S ˆ *n,2(T ˆ) n ) e2, e4, and v2 (38) The corrective function f for the characteristic energy Λ is given by the mixing rule of the interaction between the unlike ions, which is similar to eq 6b. 2
f(zCzAξ) )
xlxmpClpAmf(zClzAmξlm)/ ∑∑ l,m)1 2
xlxmpClpAm ∑∑ l,m)1
(39)
The perturbation terms are shown in Table 2. The perturbation terms for the uniunivalent mixtures are given by eqs D6-D16, which were determined by Tada et al. (1992), those for the diunivalent mixtures are eqs D17-D29, and those for the uniunivalent-diunivalent mixtures are eqs D30-D37. Figures 6-9 show the mixing properties calculated with eqs 32-35 compared to the observed ones at the mole fraction x ) 0.5 or near the fraction at a temperature for each mixture. Parts a of the figures are for the diunivalent salt mixtures, and parts b are for the uniunivalent-diunivalent salt mixtures. For some
1914 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Table 3. Standard Deviations
Figure 8. Comparison of mixing electrical conductivity ∆κcal calculated with Dd and Dξ to the observed ones ∆κobs at x = 0.5 for (a) diunivalent mixtures and (b) uniunivalent-diunivalent mixtures. The keys are the same as in Figure 6.
Figure 9. Comparison of mixing viscosity ∆ηcal calculated with Dd and Dξ to the observed ones ∆ηobs at x = 0.5 for (a) diunivalent mixtures and (b) uniunivalent-diunivalent mixtures. The keys are the same as in Figure 6.
mixtures the calculated and observed mixing properties do not agree well. ∆Vcalc of MgCl2-CaCl2 and ZnCl2-
mixing property
diunivalent mixture
uniunivalentdiunivalent mixture
∆V, cm3/mol ∆σ, dyn/cm ∆κ, S/cm ∆η, mPa‚s
0.387 0.269 0.243 0.0328
0.755 5.20 0.226 0.435
SnCl2 have an opposite sign to the ∆Vobs in Figure 6a (cf. Figure 10b) and ∆Vcalc of SrCl2-KCl and CdCl2LiCl in Figure 6b (cf. Figure 10d). ∆σcalc of MgCl2-LiCl, MgCl2-RbCl, and SrCl2-CsCl are underestimated in Figure 7b (cf. Figure 11d). ∆κcalc of CaCl2-SrCl2 and CdI2-KI are overestimated in Figure 8a (cf. Figure 12b) and Figure 8b (cf. Figure 12d). ∆ηcalc of MgCl2-NaCl has an opposite sign to ∆ηobs in Figure 9b (cf. Figure 13d). The reason for the disagreement in the surface tension of the mixtures that include MgCl2 and in the electrical conductivity of the mixture that include CdI2 may be that the surface tension of pure MgCl2 and the electrical conductivity of pure CdI2 are not correlated well in the corresponding states as shown in Figures 3 and 4. The reason for the disagreement of the other mixture is not clear at the present time. However, it can be said that the calculated and observed mixing properties satisfactorily agree on the whole. Table 3 shows root mean square deviations of the mixing properties. Because the values of the mixing properties for some salts are so small, root mean square deviations normalized with such small values would be large and can be misleading. Thus, the deviations in Table 3 are not normalized. Figures 10-13 show the mixing properties of some representative molten salt mixtures. Parts a and b of
Figure 10. Mixing molar volume of representative (a and b) diunivalent mixtures and (c and d) uniunivalent-diunivalent mixtures.
Figure 11. Mixing surface tension of representative (a and b) diunivalent mixtures and (c and d) uniunivalent-diunivalent mixtures.
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1915
Figure 12. Mixing electrical conductivity of representative (a and b) diunivalent mixtures and (c and d) uniunivalent-diunivalent mixtures.
Figure 13. Mixing viscosity of representative (a and b) diunivalent mixtures and (c and d) uniunivalent-diunivalent mixtures.
the figures are for the diunivalent mixtures, and parts c and d are for the uniunivalent-diunivalent mixtures. The solid line shows results calculated with the secondorder perturbation terms with respect to the ionic size difference Dd and the valence difference Dξ. The broken line shows results calculated with Dd and without Dξ term. The mixing properties calculated with Dd and Dξ agree well with the observed ones in parts a and c and do not agree well in parts b and d except for the mixing surface tension and viscosity of the diunivalent salt mixtures, the calculated values of which agree satisfactorily with the observed ones in Figures 11b and 13b. The valence difference perturbation term Dξ improves much the calculated mixing properties as shown in parts a and c where good agreement is obtained. The Dξ term contributes little to the calculated results as shown in parts b and d where good agreement is not obtained. The features of the equations for the mixing properties eqs 32-35 are that the equations (1) can be used for the estimation of the mixing properties of unsymmetrical (diunivalent and uniunivalent-diunivalent) salt mixtures as well as symmetrical (uniunivalent) salt mixtures, (2) have only the pair potential parameters between the unlike ions ψij, Fij, ξij, and ζij and the mass of the component ions mi, and (3) do not need any binary interaction parameters. 4. Conclusion Pair potential parameters of diunivalent salts, alkali earth metal, amphoteric metal, and transition metal halides were determined such that the molar volume, the vapor pressure, and the surface tension of the pure
salts were correlated in the corresponding states. Corresponding states equations for the molar volume, vapor pressure, surface tension, electrical conductivity, and viscosity of the pure molten salts and their mixtures were derived from the perturbation expansions of the Helmholtz free energy and the transport properties with the four pair potential parameters between the unlike ions and the ionic mass. Equations for the mixing molar volume, surface tension, electrical conductivity, and viscosity of the diunivalent molten salt mixtures and the uniunivalent-diunivalent molten salt mixtures were obtained from the corresponding states equations with satisfactory agreement to the observed values. The presented equations for the mixing properties had no binary interaction parameters. Nomenclature A ) anion A ) configuration Helmholtz free energy, dynamical quantity A˙ (t) ) time derivative of dynamical quantity A(t) C ) cation C ) coefficient of the second-order perturbation term with respect to ionic size difference for thermodynamic properties Dd ) parameter of characteristic length (ionic size) difference defined by eq 13 Dξ ) parameter of difference of product of valences and effective Coulomb potential parameter defined by eq 14 d ) characteristic length (ionic size) of mixture dij ) characteristic length (ionic size) of pure salt e ) elementary charge
1916 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 f ) f bond, corrective function for transport properties G ) coefficient of the second-order perturbation term with respect to difference of product of valences and effective Coulomb potential parameter g3 ) three body distribution function gj 3 ) constant value of g3 near rlm ) dlm and rln ) dln K ) transport property k ) Boltzmann constant mAl ) mass of anion in salt l mCl ) mass of cation in salt l m ˜ ) characteristic mass of mixture defined by eq 24 m ˜ l ) characteristic mass of pure salt l defined by eq 36 NA0 ) Avogadro’s number N ) total number of ions in a system p ) vapor pressure pAl ) stoichiometric number of anion in a component salt l pCl ) stoichiometric number of cation in a component salt l Q ˆ ) reduced coefficient of the perturbed term with respect to the mass difference and the difference of the softsphere potential in the perturbation expansion of the transport property Q ˆ ξn ) reduced coefficient of the perturbed term with respect to the mass difference and the difference in the effective Coulomb potential in the perturbation expansion of the transport property Q ˆ Fk1...kn ) reduced coefficient of the perturbed term with respect to the mass difference and the difference of the soft-sphere potential in the perturbation expansion of the transport property R ˆ ) coefficient of the second-order perturbation term with respect to ionic size difference for transport properties r ) ionic distance S ˆ n ) reduced coefficient of the perturbed term with respect to the mass difference in the perturbation expansion of the transport property T ) temperature T ˆ ) kTd/(zCzAξe2), reduced temperature of mixture T ˆ l ) kTdll/(zClzAlξlle2), reduced temperature of pure salt l t ) time V ) molar volume Xl ) physical property of pure salt l XM ) physical property of mixture xl ) mole fraction of salt l zi ) valence of the ith ion Greek Letters β ) 1/(kT) ∆ ) mixing quantity δ ) δC ) -δA ) parameter of mass difference defined by eq 26 ζlm ) characteristic parameter of pure salt η ) viscosity κ ) electrical conductivity Λ ) ψ exp(-d/F)/f µAl ) parameter of mass difference about the anion of salt l defined by eq 27 µCl ) parameter of mass difference about the cation of salt l defined by eq 27 ξ ) characteristic potential parameter of mixture ξlm ) parameter of the effective Coulomb potential of pure salt F ) characteristic potential parameter of mixture Flm ) parameter of the soft-sphere potential of pure salt σ ) surface tension φij ) pair potential between i and j ions ψ ) characteristic potential parameter of mixture ψlm ) parameter of the soft-sphere potential of pure salt ωξ ) parameter defined by eq C7 ωF ) parameter defined by eq 28 ωFk ) parameter defined by eq C5
ωψFk ) parameter defined by eq C6 Superscripts b ) blip function H ) hard-sphere molten salt with the pair potential eq 3 ˆ ) reduced form ° ) soft-sphere salt reference with the pair potential eq 2 Q ) soft-sphere salt reference with the pair potential eq B2 * ) simplified form Subscripts A ) Helmholtz free energy calc ) calculated value e ) electrical conductivity obs ) observed value p ) vapor pressure V ) molar volume v ) viscosity σ ) surface tension 0 ) reference
Appendix A The pair potential φRγij is rewritten as
φRγij ) ψ(1 + RAij) exp[-(r/F)(1 + γCij)] + (1 + RBλiνj)(zλzνξe2/r), λ, ν ) C, A (A1) where
Aij ) (ψij/ψ) - 1
(A2a)
Bλiνj ) (zλizνjξij)/(zλzνξ) - 1, λ, ν ) C, A (A2b) Cij ) (F/Fij) - 1
(A2c)
When R ) γ ) 0, φRγij ) φ°ij, whereas for R ) γ ) 1, φRγij ) φij.
φ°ij ) ψ exp(-r/F) + zλzνξe2/r, λ, ν ) C, A
(A3)
φij ) ψij exp(-r/Fij) + zλizνj ξije2/r, λ, ν ) C, A (A4) The configuration integral is defined as
∫ exp(-βΦRγN) dNrN ΦRγN ) ∑∑ φRγij i>j
(A5)
ZRγN )
(A6)
The logarithm of the configuration integral is expanded in powers of R and γ around that of the softsphere reference system:
(
ln ZRγN ) ln Z°N + R
(
) (
∂ ln ZRγN ∂R
) (
2 Rγ R2 ∂ ln Z N 2 ∂R2
+
0
+γ
0
)
2 Rγ γ2 ∂ ln Z N 2 ∂γ2
(
Rγ
)
∂ ln ZRγN ∂γ
0
+
0
+
)
∂2 ln ZRγN ∂R ∂γ
+ ... (A7) 0
The superscript ° and subscript 0 in eq A7 mean R ) γ ) 0. The second term in the right-hand side (rhs) of eq A7 can be written as follows
(
)
∂ln ZRγN ∂R
β
∫
)Z°N
0
β Z°N
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1917
( )
Equation A14 means
∂ΦRγ ∂R
N
exp(-βΦ°N) d rN )
0
ij
(A9)
From assumption 1 in the text, the terms of the repulsive core potential between like-charged ions are neglected. Integrating over 3(N - 2) coordinates gives eq A10 rewritten as follows.
(
∂ ln Z
V NA02β
Rγ
)
Appendix B The pair potential φRij is rewritten as
φRij ) ψij exp(-r/Fij) + (1 + RBλiνj)(zλzνξ e2/r), λ, ν ) C, A (B1) where Bλiνj is given by eq A2b. When R ) 0, φRij ) φQij, whereas for R ) 1, φRij ) φij (eq 1 or eq A4).
φQij ) ψij exp(-r/Fij) + zλzνξ e2/r, λ, ν ) C, A (B2) The configuration integral ZRN is defined as
N
∂R
∫ exp(-βΦRN) dNr ΦRN ) ∑∑ φRij i>j
ZRN )
) 0 2
∫
ψ exp(-r/F)g°CA(r) dr
2
∑∑ l,m)1
xlxmpClpAmAlm + ∑∑ l,m)1
∫
xlxmpClpAmBClAm (zCzAξe2/r)g°CA(r) dr (A10)
where g°CA is the radical distribution functions between the cation and the anion of the soft-sphere reference. By using the characteristic potential parameters ψ and ξ, which are expressed by eqs 6a and 6b in the text, 2 respectively, ∑∑l,m)1 xlxmpClpAmXlm ) 0 (X ) A or B) and eq A10 is reduced to the following equation:
(
)
∂ ln ZRγN ∂R
0
)0
(A11)
(
∂ ln Z
V NA02β
∂R
)
(B4)
(
) (
∂ ln ZRN ∂R
ln ZRN ) ln ZQN + R
+
0
)
2 R R2 ∂ ln Z N 2 ∂R2
+ ...
0
(B5)
The subscript 0 in eq B5 means R ) 0. The second term in the rhs of eq B5 can be written by eq B6 in a similar manner to eq A10 in Appendix A.
(
)
NA02β
∂ ln ZRN ∂R
∫(zCzAξ e2/r)gQCA(r)
)V
0
2
dr
xlxmpClpAmBClAm ∑∑ l,m)1
(B6)
where gQCA is the radial distribution functions between the cation and the anion of the system with the pair potential φQij. Equation B6 vanishes with eq 6b in the text.
N
)
0
(
2
∫(ψr/F) exp(-r/F)g°CA(r) dr ∑∑ xlxmpClpAmClm (A12) In a similar manner to eqs A10 and A11, 2 xlxmpClpAmClm ) 0, and eq A12 is reduced to eq ∑∑l,m)1 A13 by using the characteristic parameter F, eq 6c.
)
∂ ln ZRγN ∂γ
0
)0
(A13)
Since the two or higher order terms of R and γ in eq A7 are neglected from assumption 2 in the text, using eqs A11 and A13 and letting R ) γ ) 1 reduce eq A7 to eq A14.
ln ZN ) ln Z°N
(A14)
)
∂ ln ZRN ∂R
l,m)1
(
(B3)
The logarithm of the configuration integral is expanded in powers of R around that of the system in which ions interact through the pair potential φQij.
Since the third term in the rhs of eq A7 has only the interactions of the repulsive core potential, it can be expressed as Rγ
(A15)
which is eq 7 in the text.
∫ ∑∑ [Aijψ exp(-r/F) +
Φ°N )
-A ˆ ) -βA ) -βA° ) -A ˆ°
)0
0
(B7)
The third term in the rhs of eq B5 can be written by eq B8
(
) [ ∫( )
∂2 ln ZRN ∂R2
)-
0
∫
β ZQN
β ZQN
( )
∂ΦRN ∂R
∂2ΦRN 2
∂R 2
β ZQN
0
0
2
-
exp(-βΦQN) dNr +
( )
∫
]
exp(-βΦQN) dNr
∂ΦRN ∂R
0
2
exp(-βΦQN) dNr (B8)
The first term in the rhs of eq B8 becomes the square of eq B6 and vanishes. The second term in the rhs of eq B8 vanishes because the pair potential φRij, eq B1, is
1918 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
a linear function of R. The third term in the rhs of eq B8 is rewritten by eq B9
G ˆ ′′A(T ˆ) ) V2
third term of eq B8 ) NA02β2 V
∫
( ) zCzA ξ e2
Dξ )
[
V2
rCArCA′
xlxmxnpClpAmpAnBClAmBClAn + ∑∑∑ l,m,n)1 (zCzAξe2)2 gQ3(AC,AC′) drAC drAC′ × rACrAC′
∫
xlxmxnpAlpCmpCnBCmAlBCnAl] + ∑∑∑ l,m,n)1
[∫
2
V
rCArC′A′
2
gQ4(CA,C′A′) drCA drC′A′ × 2
∫
gQ4(AC,A′C′) drAC drA′C′ ×
rACrA′C′
2
∑∑ k,l)1
]
2
xkxlpAkpClBClAk
xmxnpAmpCnBCnAm ∑∑ m,n)1
(B9)
where g3 is a three-body distribution function which is proportional to the probability density that ions 1, 2, and 3 exist at positions r1, r2, and r3, respectively, and interact via e bonds, exp(βφ12) and exp(βφ13), and g4 is a four-body distribution function which is proportional to the probability density that ions 1, 2, 3, and 4 exist at positions r1, r2, r3, and r4, respectively, and interact via e bonds, exp(-βφ12) and exp(-βφ34). The second square-bracketed term in the rhs of eq B9 vanishes due to the definition of BClAm, eq A2b. Assumption 4. The integrals in eq B9 do not diverge due to the effect of screening and are given by functions only of the reduced temperature T ˆ. With assumption 4, the third term of eq B8 becomes
ˆ )Dξ + G ˆ ′A(T ˆ )D′ξ + third term of eq B8 ) G ˆ A(T ˆ )D′′ξ (B10) G ˆ ′′A(T G ˆ A(T ˆ) ) G ˆ ′A(T ˆ) ) NA03β2 V2
NA02β2 V
∫
(
∫
(
)
zCzAξe2 r
)
zCzAξe2 rCArCA′
D′ξ )
xlxmxnpClpAmpAnBClAmBClAn ∑∑∑ l,m,n)1
D′′ξ )
xlxmxnpAlpCmpCnBCmAlBCnAl ∑∑∑ l,m,n)1
(B15)
2
gQCA(r) dr
(B16)
Thus, eq B8 is reduced to
(
)
∂2 ln ZRN ∂R2
0
)G ˆ A(T ˆ )Dξ + G ˆ ′A(T ˆ )D′ξ + G ˆ ′′A(T ˆ )D′′ξ (B17)
ˆ A(T ˆ )Dξ + G ˆ ′A(T ˆ )D′ξ + G ˆ ′′A(T ˆ )D′′ξ ln ZN ) ln ZQN + G (B18)
xkxlpCkpAlBCkAl ∑∑ xmxnpCmpAnBCmAn + ∑∑ k,l)1 m,n)1 (zCzAξe2)2
(B14)
Substitution of eqs B7 and B17 into eq B5 with R ) 1 and assumption 3 in the text gives eq B18.
2
(zCzAξe2)2
xlxmpClpAmBClAm2 ∑∑ l,m)1
2
gQ3(CA,CA′) drCA drCA′ ×
2
NA03β2
gQ3(AC,AC′) drAC drAC′ (B13)
2
xlxmpClpAmBClAm2 + ∑∑ l,m)1
∫
2
2
gQCA(r) dr ×
r
(zCzAξe2)2
)
zCzAξe2 rACrAC′
∫
2
2
NA03β2
(
NA03β2
(B11)
2
gQ3(CA,CA′) drCA drCA′ (B12)
ln ZQN, the first term in the rhs of eq B18, is expanded around the configurational integral of the hard-sphere molten salt reference ln ZHN in a similar way to the uniunivalent molten salt mixtures (Tada et al., 1990a, 1992). For diunivalent molten salts the numbers of cation and anion pCl and pAm must be taken into account. ln ZQN is expressed in terms of diagrams (Morita and Hiroike, 1961) to be the sum of all simple irreducible diagrams composed of two or more black density circles and Mayer f bonds. The Mayer f bond is divided into two parts:
f ) fH + fb
(B19)
f ) exp(-βφ) - 1
(B20)
fH ) exp(-βφH) - 1
(B21)
Substitution of eq B19 into ln ZQN diagrams yields
ln ZQN ) ln ZHN + [sum of simple irreducible diagrams involving only one fb bond] + [sum of simple irreducible diagrams involving two fb bonds which are connected by a black circle] + [sum of simple irreducible diagrams involving two fb bonds which are connected at least by two black circles and an fH bond] + [sum of simple irreducible diagrams involving at least three fb bonds] (B22) The characteristic separation distance d is chosen such that the first square-bracketed term vanishes.
first square-bracketed term of eq B22 ) 0 (B23) Equation B23 can be described by integral of the radial distribution functions and fb bonds and is reduced to eq B24 in a similar manner to the report of Tada et al. (1990a) except that only the diagrams involving fb bonds between unlike ions are treated from assumption 1 in
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1919
the text. 2
2
xlxmpClpAmdlm3)/(∑∑ xlxmpClpAm) ∑∑ l,m)1 l,m)1
d3 ) (
(B24)
Equation C1 is expanded with the mass difference of the component ions and the differences of the potential parameters (Tada et al., 1990b) ∞
which is eq 9 in the text. The second square-bracketed term of eq B22 is expressed with g3 and is reduced to eq B25 in a similar manner to the report of Tada et al. (1992) except that only the diagrams involving fb bonds between unlike ions are treated from assumption 1 in the text.
K ˆ )K ˆ0 +
∑ Sˆ nδn + n)1 ∞
(
∞
ˆ Fk ...k ωFk ...ωFk ∑ ∑ ... ∑ Q n)1 k , ... , k )1 ∞
1
1
where
C ˆ A(T ˆ ) ) 4πNA03gj 3d6
(B26)
1
3
3
n
1
n
)
+Q ˆ ξnωξn (C4)
n
2
ωFk )
xlxmxn [pClpAmpAn((dlm3/d3) ∑∑∑ l,m,n)1 3
+ n
where δ is defined by eqs 25, 26, and 27 in the text. ωFk, ωψFk, and ωξ are defined as follows:
2
Dd )
1
ˆ Fk ...k ωψFk ...ωψFk ∑ ... ∑ Q k , ... , k )1 1
ˆ )Dd second square-bracketed term in eq B22 ) C ˆ A(T (B25)
n
n
xlxmpClpAm((F/Flm) - 1)k(µCl + µAm) ∑∑ l,m)1
(C5)
2
ωψFk )
3
1)((dln /d ) - 1) + pAlpCmpCn((dml /d ) 1)((dnl3/d3) - 1)] (B27) Here gj 3 is a constant value of g3 near rlm ) dlm and rln ) dln from assumption 5. Assumption 5. gj 3 does not depend on the component salts in the mixture and depends only on the reduced temperature T ˆ ) kTd/(zCzAξe2). The third square-bracketed term of eq B22 is expressed with g4 and vanishes when the characteristic length d is chosen such that eq B24 or eq 9 in the text is satisfied (Tada et al., 1992).
xlxmpClpAm((ψlm/ψ) - 1)((F/Flm) - 1)k × ∑∑ l,m)1 (µCl + µAm) (C6) 2
ωξ )
xlxmpClpAm[(zClzAmξlm/(zCzAξ)) - 1](µCl + ∑∑ l,m)1 µAm) (C7)
Substitution of eqs B23, B25, and B28 into eq B22 with assumption 3 in the text gives
K ˆ 0 is the reduced transport property for the reference system, whose ions have the unique mass m ˜ and interact through the reference potential φ°ij. The parameters ωFk, ωψFk, and ωξ contain the number parameters of cation and anion of the component salt pCl and pAm so that eq C4 can be applied to the diunivalent salts as well as the uniunivalent salts. The reduced transport property K ˆ is rewritten as
ln ZQN ) ln ZHN + C ˆ A(T ˆ )Dd
K ˆ )K ˆ *V ˆT ˆ zCzAξe2/(Λd)
(C8)
K ˆ * ≡ Kd2Λ2/[A2(m ˜ Λ)1/2]
(C9)
third square-bracketed term of eq B22 ) 0 (B28)
(B29)
Substitution of eqs B29 into eq B18 gives H
ln ZN ) ln Z
N
+C ˆ A(T ˆ )Dd + G ˆ A(T ˆ )Dξ + G ˆ ′A(T ˆ )D′ξ + ˆ )D′′ξ (B30) G ˆ ′′A(T
When eq B30 is applied to the molar volume and surface tension of the mixture, the fourth and fifth terms in the rhs of eq B30 contributed little to the mixing properties. Thus, eq B30 is approximated to be reduced to the following equation, which is equivalent to eq 11 in the text.
ln ZN ) ln ZHN + C ˆ A(T ˆ )Dd + G ˆ A(T ˆ )Dξ
(B31)
Appendix C
∞
Equation 21 is rewritten in a reduced form
K ˆ )
KVkT Λ 1/2 ˜ A2d m
( )
∫0∞〈Aˆ˙ (0) Aˆ˙ (tˆ)〉 dtˆ
V ˆ is expressed by eq 18 in the text. When V ˆ is discarded from K ˆ , account must be taken with respect to the ionic size difference Dd and the ionic valence difference Dξ. The Coulomb potential would play a minor role in the transport properties due to the smearing effect, as shown by Rice (1962). Thus, the scaled Coulomb potential parameter zCzAξe2/(Λd) has a minor effect on the transport properties. The reduced transport property K ˆ can be replaced by the simple one K ˆ *, by discarding zCzAξe2/(Λd), V ˆ , and T ˆ in K and adding the second-order perturbation terms Dd and Dξ in eq C4. 2
(C1)
∞
1/2
(C2)
∞
1
1
1
n
n
n
ˆ *Fk ...knωψFk ...ωψFk ∑ ... ∑ Q k , ... , k )1 1
(C3)
(
S ˆ *nδn + ∑ n)1
∑ ∑ ... ∑ Q*Fk ...k ωFk ...ωFk n)1 k , ... , k )1 ∞
ˆ˙ (tˆ) ) A˙ (t)/A A
2
K ˆ * ) Kd Λ /[A (m ˜ Λ) ] ) K ˆ *0 +
where
ˆt ) (t/d)(Λ/m ˜ )1/2
2
1
n
1
+
n
)
+Q ˆ *ξnωξn + R ˆ *Dd + G ˆ *Dξ (C10)
1920 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
ωFkn, ωψFkn, ωξ, Dd, and Dξ are zero in the pure molten salts from their definitions. For the uniunivalent salts the odd order terms with respect to the mass difference δ2n-1 vanish due to the valence symmetry (Tada et al., 1990b). The observed data of the pure alkali halides showed that K ˆ * does not depend on terms of order higher than δ4 from the plot of K ˆ * against δ2 at any reduced temperatures. The observed data of the alkali halide mixtures showed that K ˆ * does not depend on terms of order higher than ωF2 (ωF ) ωF1) and any terms which include ωFn (n g 2), ωψFn, and ωξ from the plot of K ˆ* - S ˆ *2δ2 - S ˆ *4δ4 against any of the ωFn (n g 2), ωψFn, and ωξ terms. Thus, for simplicity eq C10 is reduced to eq C11 both for symmetry (uniunivalent) salt mixtures and unsymmetry (diunivalent and uniunivalent-diunivalent) salt mixtures
K ˆ* ) K ˆ *0 + S ˆ *2δ2 + S ˆ *4δ4 + Q ˆ *1ωF + Q ˆ *2ωF2 + R ˆ *Dd + G ˆ *Dξ (C11) where ωF ) ωF1, Q ˆ *1 ) Q ˆ *F1, and Q ˆ *2 ) Q ˆ *F2. Literature Cited Andersen, H. C.; Weeks, J. D.; Chandler, D. Relationship between the Hard-Sphere Fluid and Fluids with Realistic Repulsive Forces. Phys. Rev. 1971, A4, 1597. Davis, H. T. Theory of Heats of Mixing of Certain ChargeUnsymmetrical Fused Salts. J. Chem. Phys. 1964, 41, 2761. Harada, M.; Tanigaki, M.; Tada, Y. Law of Corresponding States of Uniunivalent Molten Salts. Ind. Eng. Chem. Fundam. 1983, 22, 116-121. Hersh, L. S.; Kleppa, O. J. Enthalpies of Mixing in Some Binary Liquids Halide Mixtures. J. Chem. Phys. 1965, 42, 1309. Janz, G. J. Molten Salts Handbook; Academic Press: New York, 1967. Janz, G. J.; Gardner, G. L.; Krebs, U.; Tomkins, R. P. T. Molten salts: Volume 4, Part 1, Fluorides and Mixtures. Electrical Conductance, Density, Viscosity, and Surface Tension Data. J. Phys. Chem. Ref. Data 1974, 3, 1. Janz, G. J.; Tomkins, R. P. T.; Allen, C. B.; Downey, J. R., Jr.; Gardner, G. L.; Krebs, U.; Singer, S. K. Molten salts: Volume 4, Part 2, Chlorides and Mixtures. Electrical Conductance, Density, Viscosity, and Surface Tension Data. J. Phys. Chem. Ref. Data 1975, 4, 871. Janz, G. J.; Tomkins, R. P. T.; Allen, C. B.; Downey, J. R., Jr.; Singer, S. K. Molten salts: Volume 4, Part 3, Bromides and Mixtures; Iodides and Mixtures. Electrical Conductance, Density, Viscosity, and Surface Tension Data. J. Phys. Chem. Ref. Data 1977, 4, 409. Janz, G. J.; Tomkins, R. P. T.; Allen, C. B. Molten salts: Volume 4, Part 4, Mixed Halide Melts. Electrical Conductance, Density, Viscosity, and Surface Tension Data. J. Phys. Chem. Ref. Data 1979, 8, 125.
Kleppa, O. J.; Hersh, L. S. Heats of Mixing in Liquid Alkali Nitrate Systems. J. Chem. Phys. 1961, 34, 351. Luks, K. D.; Davis, H. T. Recent Statistical Mechanical Theories of the Thermodynamic Properties of Molten Salts. Ind. Eng. Chem. Fundam. 1967, 6, 194. McCarty, F. G.; Hersh, L. S.; Kleppa, O. J. Note on the Heats of Mixing in Charge-Unsymmetrical Fused Salt Solutions. J. Chem. Phys. 1964, 41, 1522. McCarty, F. G.; Kleppa, O. J. Thermochemistry of the Alkali Chloride-Lead Chloride Liquid Mixtures. J. Phys. Chem. 1964, 68, 3846. Morita, M.; Hiroike, K. A New Approach of the Theory of Classical Fluids. III. General Treatment of Classical Systems. Prog. Theor. Phys. 1961, 25, 537. Pitzer, K. S. Corresponding States for Perfect Liquids. J. Chem. Phys. 1939, 7, 583. Reiss, H.; Mayer, S. W.; Katz, J. Law of Corresponding States for Fused Salts. J. Chem. Phys. 1961, 35, 820. Reiss, H.; Katz, J.; Kleppa, O. J. Theory of the Heats of Mixing of Certain Fused Salts. J. Chem. Phys. 1962, 36, 144. Rice, S. A. Kinetic Theory of Ideal Ionic Melts. Trans. Faraday Soc. 1962, 58, 499. Tada, Y.; Hiraoka, S.; Uemura, T.; Harada, M. Corresponding States Correlation of Transport Properties of Uniunivalent Molten Salts. Ind. Eng. Chem. Res. 1988, 27, 1042. Tada, Y.; Hiraoka, S.; Uemura, T.; Harada, M. Law of Corresponding States of Uniunivalent Molten Salt Mixtures. 1. Mixing Rule of Pair Potential Parameters. Ind. Eng. Chem. Res. 1990a, 29, 1509. Tada, Y.; Hiraoka, S.; Uemura, T.; Harada, M. Law of Corresponding States of Uniunivalent Molten Salt Mixtures. 2. Transport Properties. Ind. Eng. Chem. Res. 1990b, 29, 1516. Tada, Y.; Hiraoka, S.; Katsumura, Y.; Yamada, I. Effects of Ionic Size Difference on Thermodynamic and Transport Properties of Uniunivalent Molten Salt Mixtures. Ind. Eng. Chem. Res. 1992, 31, 2010. Tada, Y.; Hiraoka, S.; Katsumura, Y.; Park, C.-H. Simplified Equations for Mixing Properties of Molten Alkali Halides Based on the Law of Corresponding States. Ind. Eng. Chem. Res. 1993, 32, 2873. Tada, Y.; Hiraoka, S.; Achiwa, T.; Koh, S.-T. Mixing Properties of Molten Alkali Nitrates Based on the Law of Corresponding States. Ind. Eng. Chem. Res. 1995, 34, 1461. Tosi, M. P.; Fumi, F. G. Ionic Sizes and Born Repulsive Parameters in the NaCl-type Alkali-Halides. II. The Generalized HugginsMayer Form. J. Phys. Chem. Solids 1964, 25, 45. Young, R. E.; O’Connell, J. P. An Empirical Corresponding States Correlation of Densities and Transport Properties of 1-1 Alkali Metal Molten Salts. Ind. Eng. Fundam. 1971, 10, 418.
Received for review September 9, 1996 Revised manuscript received January 6, 1997 Accepted January 17, 1997X IE9605544 X Abstract published in Advance ACS Abstracts, March 1, 1997.
