Ind. Eng. Chem. Res. 1995,34, 1461-1470
1461
Mixing Properties of Molten Alkali Nitrates Based on the Law of Corresponding States Yutaka Tada,* Setsuro Hiraoka, Takashi Achiwa, and Seung-Tae Koht Department of Applied Chemistry, Nagoya Institute of Technology, Nagoya 466, Japan
The law of corresponding states of molten alkali nitrates and their mixtures was developed. Pair potential parameters of the pure salts were determined such that the observed molar volume and surface tension were correlated in the corresponding states. Equations for the mixing enthalpy, 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. Estimation of the mixing properties with two characteristic parameters, characteristic molar volume and temperature, was also discussed. 1. Introduction
Statistical mechanical theories of the thermodynamic properties of molten salts were discussed by Lucks and Davis (1967). They showed that Reiss et al. (1961) developed the theory of corresponding states for symmetric and unsymmetric pure molten salts with a simple ionic model through Pitzer's dimensional analysis (Pitzer, 1939). In the ionic model, the pair potential is Coulomb interaction between like ions and is the sum of Coulomb interaction and hard-sphere repulsion between unlike ions. Polarization and dispersion effects are accounted for by an effective dielectric constant. 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. They expanded the excess free energy of mixtures around that of the reference salt, component salt 2, with respect to the difference between the coupling parameters and the difference between scaling parameters for length to give first-order and second-order approximations. Hersh and Kleppa (1965) showed that the heat of mixing of alkali nitrate mixtures and alkali halide mixtures, in which the cations are not highly polarized, correlated well with the coupling parameters of the nonCoulombic interactions and the difference between the length parameters. In the works mentioned above, the theory of corresponding states for transport properties of pure molten salts was not developed and the mixing molar volume, surface tension, and transport properties of molten salts were not predicted. Pair potential for molten alkali halides was simplified to the sum of soft-sphere repulsion and effective Coulomb potential which incorporated effects of polarization and dispersion by Harada et al. (1983). They scaled the soft-sphere potential 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. They showed that the thermodynamic properties of pure molten alkali halides were correlated in the corresponding states by using the hard-sphere diameter and the effective Coulomb potential parameter as the characteristic parameters. + Present address: TOY0 Engineering Co. Ltd, Tokyo, Japan.
Tada et al. (1988) used the simplified pair potential to show that the transport properties of the pure molten salts were correlated in the corresponding states with the potential parameters and 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) applied the simplified pair potential to the mixtures of molten alkali halides. The corresponding states equations of the mixtures were obtained with the characteristic potential parameters and masses which were determined from perturbation expansions of the Helmholtz free energy and the transport property of the mixtures. The mixing properties were estimated well by using the corresponding states equations. The aim of this paper is to obtain the equations for the mixing properties of molten alkali nitrates without using binary interaction parameters. The simplified pair potential is applied to the mixture, and the thermodynamic and transport properties are correlated in the corresponding states. The equations for the mixing properties are obtained from the corresponding states equations. It is also shown that the mixing properties are estimated with two parameters, characteristic molar volume and temperature. 2. Equations of Corresponding States
A binary mixture of molten alkali nitrates, salt 1and salt 2, is considered. Salt 1 comprises N1 cations and N I anions, and salt 2 comprises N2 cations and N2 anions. It is postulated that the pair potential of 1 and m ions is expressed by eq 1, which was proposed by Harada et al. (1983) for alkali halides.
where z1 and e are the valence of ion 1 and elementary charge, respectively. vim and elm are parameters of core repulsive potential, and Elm is a parameter of effective Coulomb potential. It is also postulated that a parameter between likecharged ions of salt i (i = 1 , 2 ) is equal to that between unlike-charged ions of the salt and that a parameter between like-charged ions of different salts 1 and 2 is equal to the arithmetic mean of parameters of unlikecharged ions between the different salts (Tada et al., 1990a). For instance, in a mixture of NaNO3 (salt 1) and KNo3 (salt 2)
0888-5885/95/2634-1461$09.00/00 1995 American Chemical Society
1462 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995
zj,k=l
[(ddd)3- 1][(djk/d)3 - 11 qNaK
= qNO3(1)N0&2)= (VNaNO, -k VKNO3)l2
+ [(dik/d)3- l][(djk/d)3111 (10)
(2)
where subscript Nodi) means nitrate ion in salt i (i = 1, 2). Helmholtz free energy of the mixture is expanded with the difference between the core repulsive potentials around a hypothetical pure molten salt which has softsphere Coulomb potential with core repulsive potential parameters ly and e and a n effective Coulomb potential parameter 6. The Helmholtz free energy of the mixture is expanded with the difference between the core repulsive potentials around another hypothetical pure molten salt which has hard-sphere Coulomb potential with the hard-sphere diameter d and the effective Coulomb potential parameter 5. The potential parameters ly, e, (, and d are determined by equating the first-order terms of the perturbation expansions to 0 and are given by the following mixing rules (Tada et al., 1990a):
a parameter of the ionic size differences, and C.dT)Dd is the second-order perturbation term. Equation 8 means that the thermodynamic properties follow the law of corresponding states with the error of the perturbation terms higher than second order, using the characteristic length d and the characteristic potential parameter t. The thermodynamic properties which are derived from the Helmholtz free energy, i.e., the enthalpy, molar volume, and surface tension, of the molten salt mixtures along the saturation curves can be related to the following universal functions, if density dependence of: th? d value is neglected (Tada et al., 1992). Here, HH, P, and aHare the reference terms of
Dd i s
A = H/(kT)= AH(!?)+ e d f i D d
(11)
Q = v/(Nd3)= p(n+ e)V(fiDd
(12)
P(n+ eg(@Dd
(13)
6 = 0d3/(te2)=
0
the hypothetical hard-sphere molten salt. Collective transport properties, i.e., electrical conductivity and viscosity, of the mixture are reduced with a Characteristic mass f i and the characteristic potential parameters v, e, and t and are expanded with the mass difference of the component ions and the ionic size difference around the hypothetical soft-sphere molten salt, whose ions have the unique mass f i , The transport properties along the saturation curves can be related to the following universal functions, if density dependence of the d value is neglected (Tada et al., 1992). m
ij = l
where xi means mole fraction of salt z and I&, e?, and t~mean potential parameters between the cation in salt i and the anion in saltj. dij is a characteristic length of a pure salt which consists of a cation from salt i and an anion from saltj. dij is determined by the perturbation of the pure salt and is expressed by eq 7 (Harada et al., 1983). A fourth parameter I;u, which incorporates the
dJeij = 0.4069
+ cSe2,*(nS2" + n=l
14)
15)
+ 0.9075 ln(q4kT) + 6.042 x 10-7(lydkT) (7)
effects caused by the difference in size of the cation in salt i and the anion in salt j, was involved in the dy equation of Harada et al. (1983) for molten alkali halides. However, the thermodynamic and transport properties of molten alkali nitrates are correlated in the corresponding states without cy as shown below; therefore, eq 7 does not need
