Ind. Eng. Chem. Res. 1990,29, 1509-1516
(ii) The effects of particle size and solid loading in baffled columns on liquid- and solid-phase mixing times were found to be analogous to those observed in unbaffled columns. Acknowledgment The present work was supported by a grant from the Board of Research in Nuclear Sciences, Department of Atomic Energy, Government of India (Project 35/4/86-G). Nomenclature a = constant in eq 1 A = distance between column axis and the point of maximum vorticity, m B = point of maximum vorticity do = diameter of the central hole, m d, = particle diameter, pm DL= liquid-phase dispersion coefficient, m2 s-l HB = baffle spacing, m H, = clear liquid height, m HD = height of dispersion, m T = column diameter, m VG= superficial gas velocity, m s-l Greek Letters CG = fractional gas holdup t S = fractional solid holdup
8 = liquid-phase mixing time for unbaffled bubble column, S
8s = liquid-phase mixing time in the presence of baffles, s Os = solid-phase mixing time for unbaffled bubble column, S
8 s = ~ solid-phase mixing time in the presence of baffles, s
Literature Cited Blenke, H. In Biotechnology; Brauer, H., Ed.; VCH Weinhein, 1985; VOl. 11.
1509
Deckwer, W. D.; Graeser, V.; Serplemen, Y.; Langemann, H. Zones of Different Mixing in the Liquid Phase of Bubble Columns. Chem. Eng. Sci. 1973,28, 1223-1225. Doraiswamy, L. K.; Sharma, M. M. Heterogeneous ReactiomAnalysis, Examples and Reactor Design; vol. 11, Wiley-Interscience: New York, 1984; Vol. 11. Field, R. W.; Davidson, J. F. Axial Dispersion in Bubble Columns. Trans. Inst. Chem. Eng. 1980,58, 228-236. Hackl. A,: Wurian. H. Determination of Mixing Time. Ger. Chem. Eng. 1979,2, 103-107. Joshi, J. B. Axial Mixing in Multiphase Contactors-A Unified Correlation. Trans. Inst. Chem. Ena. 1980.58, 115-165. Joshi, J. B.; Sharma, M. M. Some Design Features of Radial Baffles in Sectionalized Bubble Columns. Can. J. Chem. Eng. 1979,57, 375-377. Joshi, J. B.; Shertukde, P. V.; Godbole, S. P. Modelling of Three Phase Sparged Catalytic Reactors. Reu. Chem. Eng. 1988, 6, 72-156. Khare, A. S.; Dharwadkar, S. V.; Joshi, J. B. Solid Phase Mixing in Three Phase Sparged Reactor. J. Chem. Eng. Jpn. 1989, 22, 125-130. Murakami, Y .; Hiranono, T.; Ono, S.; Nishijima, T. Mixing Properties in Loop Reactor. J. Chem. Eng. Jpn. 1982, 15, 121-125. Pandit, A. B.; Joshi, J. B. There Phase Sparged Reactors Some Design Aspects. Reu. Chem. Eng. 1984,2, 1-84. Pandit, A. B.; Joshi, J. B. Mass and Heat Transfer Characteristics of Three Phase Sparged Reactors. Chem. Eng. Res. Des. 1986,64, 125-157. Parulekar, S. J.; Shertukde, P. V.; Joshi, J. B. Underutilization of Bubble Column Reactors Due to Desorption. Chem. Eng. Sci. 1989, 44, 543-558. Patil, V. K.; Joshi, J. B.; Sharma, M. M. Sectionalized Bubble Column; Gas-hold-up and Wall Side Solid-Liquid Mass Transfer Coefficient. Can. J. Chem. Eng. 1984, 62, 228-232. Shah, Y. T. Gas Liquid-Solid Reactor Design; McGraw-Hill: New York, 1978. Van de Vusse, J. G. A New Model for the Stirred Tank Reactor. Chem. Eng. Sci. 1962, 17, 507-521. I
Received for review April 27, 1989 Revised manuscript received January 17, 1990 Accepted January 25, 1990
Law of Corresponding States of Uniunivalent Molten Salt Mixtures. 1. Mixing Rule of Pair Potential Parameters Yutaka Tada,* S e t s u r o Hiraoka, and Tomokazu U e m u r a Department of Applied Chemistry, Nagoya Institute of Technology, Nagoya 466, J a p a n
Makoto Harada Institute of Atomic Energy, Kyoto University, Uji, Kyoto 611, J a p a n
A mixing rule for the pair potential parameters of molten alkali halide mixtures was obtained. The pair potential between the ions was simplified to soft-sphere and effective Coulomb potentials. The characteristic potential parameters of the mixtures were selected such that the Helmholtz energy of the mixture was equal to that of a hypothetical reference system. Molar volume and surface tension of the mixtures were correlated in a corresponding states correlation using the characteristic parameters. I t was also discussed how the mixing volume and surface tension were described by using the characteristic parameters. 1. Introduction
A law of corresponding states is one of useful methods for predicting physical properties of molten salt mixtures. The characteristic parameters that are used to reduce the temperature, distance between two ions, volume, and other physical properties should be determined theoretically. The pair potential of molten salt is expressed by the sum of Coulomb, core-repulsion, dipole-dipole (dispersion), induced-dipole-ion, and dipole-quadrupole interactions. O ~ S S - ~ S901 S ~2629-1509$02.50/0 /
It is important in a law of corresponding states how the pair potential is simplified and what are chosen as the characteristic parameters. Reiss et al. (1961) simplified the pair potential to the sum of Coulomb interaction and hard-sphere repulsion, which is the same for unlike-charged ions and for likecharged ions based on the assumption that the short-range repulsion between like ions contributes little to the configuration integral. They examined the corresponding 1990 American Chemical Society
1510 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
states for pure molten salts with the simplified pair potential. The softness of the core potential and the weak longrange potentials are important for understanding the properties of molten salts in a wide range of state variables. Harada et al. (1983) used the pair potential expressed by the sum of soft-sphere repulsion and effective Coulomb potential. The core repulsion was postulated to be the same for unlike ions and for like ions based on the Reiss assumption. The effective Coulomb potential incorporated effects of the dispersion and polarizability. They showed that the thermodynamic properties of pure uniunivalent molten salts were correlated in the corresponding states. As for the mixtures, Lucks and Davis (1967) modified the pair potential to give
@,,
=
a:
r 5 d,,
(la)
@,,= z , z J e 2 / r+ D,,@NC(r): r > d ,
(lb)
2. Characteristic Potential Parameters $, 5, p, and d We consider a binary mixture of uniunivalent molten salts: salt “1”and “2”. Salt 1 comprises N1 cations and Nl anions, and salt 2 comprises N2cations and N2 anions. Thus, the number of the total ions in the mixture N = 2(N1 + N 2 ) . It is postulated that the pair potential between i and j ions is expressed by a function of separation distance r: 4ij(r) =
for unlike-charged ions and
cplJ = z l z , e 2 / r + D,@NC(r):
It is shown that molar volume and surface tension of the alkali halide mixtures are correlated in corresponding states using the characteristic parameters that are the potential parameters of the soft-sphere reference system and the ion diameter of the hard-sphere reference system. It is also discussed how the mixing volume and surface tension are described by using the characteristic parameters obtained with the simplified pair potential.
r>0
(IC)
for like-charged ions, where @NC is the non-Coulombic potential, d,, the closest distance between unlike-charged ions, and D,,the coupling parameter of @Nc. The Helmholtz energy is expanded in powers of the difference of d,’s and that of D,,’s. It was shown that the observed mixing enthalpy was correlated with the difference of d,’s and the potential energy difference arising from the dispersion effect and that the contribution of the dispersion and polarizability is small for alkali halides. The expressions of the excess Helmholtz energy, however, were very complicated with unknown integrals, and the temperature dependency of d , was not evaluated. Lantelme and Turg (1979) investigated mixtures of LiBr-KBr on the basis of molecular dynamics, using the Tosi-Fumi (1964) pair potential, which consists of core repulsion, Coulomb, dipole-dipole, and dipole-quadrupole interactions. They showed that many properties of the mixtures were satisfactorily described by the Tosi-Fumi potential and that the contribution of the ionic polarization was relatively unimportant to the mixing properties of alkali halides. The aim of this paper is to obtain mixing rules of pair potential parameters of alkali halide mixtures. The pair potential is modified to the sum of core repulsion interaction and effective Coulomb interaction proposed by Harada et ai. (1983). The Helmholtz energy or logarithm of the configuration integral of a uniunivalent molten salt mixture is expanded around that of a hypothetical pure molten salt (reference system) in powers of the difference of the potential parameters. The potential parameters of the reference system, which are determined such that the first-order perturbation terms vanish, are expressed as mixing rules of the potential parameters of the pure salts. The Helmholtz energy of the mixture, which is expressed in a diagrammatic method of Morita and Hiroike (1961), is also expanded in powers of the difference of the core repulsive potentials around a hypothetical hard-sphere molten salt in which the ions interact through hard-sphere repulsion and effective Coulomb potential. The hardsphere diameter is selected such that the Helmholtz energy of the mixture is equal to that of the hard-sphere molten salt and is expressed as a mixing rule of the characteristic ionic distances of the pure component salts, which are given by a function of the pair potential parameters of the pure salts and temperature, eq 19 in the report of Harada et al. (1983)
IC.,,
exp(-r/pij)
+~~je’t[j/r
(2)
where $, and pl, are potential parameters of repulsion. The second term in eq 2 represents the Coulombic potential between i and j ions with valences z, and z , respectively. t,, is a parameter that incorporates the eflects of the dipole-dipole and induced-dipole-ion interactions and the dielectric constant. We introduce another pair potential with perturbation parameters cy and y:
14 + a(ti, - [)IT
zizje2
(3)
where 0 ICY I 1 and 0 I y I 1. Ij/, p, and .$are the characteristic potential parameters. When CY = y = 0, the potential @$? = @ O i j reduces to that of a hypothetical soft-sphere reference system, whereas for a = y = 1, the potential 47 = @i, is that of the mixture of interest. The logarithm of the configuration integral of the system with the pair potential in eq 3, In Z.,Y, is expanded in powers of cy and y around that of the soft-sphere reference system (eq A7 in Appendix A). The strong Coulomb interactions determine a locally ordered structure wherein cations are surrounded by anions and vice versa, and the short-range repulsive interactions between the like-charged ions are less important than those of the unlike-charged ions. Thus, we assume the following. Assumption 1. The parameter between the likecharged ions of salt I ( E = 1, 2) is equal to that between the unlike-charged ions of the salt, and the parameter between the like-charged ions of different salts 1 and 2 is equal to the arithmetic mean of the parameters of the unlike-charged ions between the different salts (eq A10 in Appendix A). The characteristic potential parameters $, E, and p are chosen to respectively be
where Glm, tlm, and plm are the parameters of the pair PO-
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1511 18r
tential between cation in salt 1 and anion in salt m. Using eqs 4a-c and letting Q = y = 1 with assumptions 1and 2, which is stated below, give (see Appendix A) In Z N = In Z O N (5)
I
I
I
11.8
Assumption 2. The higher order terms of Q and y of not less than 2 in the expansion of In ZgYcontribute little to In Z g . Equation 5 means that the Leduced configuration Helmholtz energy of the mixture, A PA, is equal to that of the soft-sphere reference system, Ao PAo. The pair potential of a hard-sphere molten salt is expressed as # ( r ) = m, r Id = eiej[/d, r
>d
(6)
where d is a characteristic separation distance. The reduced configuration Helmholtz energy, A, of the mixture can be expressed in terms of diagrams (Morita and Hiroike, 1961). The Mayer f bond in the diagrams can be devided into two parts, f bond for the hard-sphere reference and f b bond, which was called the blip function by Andersen et al. (1971). The reduced Helmholtz energy, A, is expanded in powers of the f b bond. The f b bond is nonzero only in a small range of the interionic potential. Thus, we assume the following. Assumption 3. The higher order perturbed terms that contain at least two f b bonds are assumed to be neglected. The characteristic separation distance is chosen such that the first perturbed diagram that contains only one f bond vanishes. This choice of d and assumption 3 give (7)
A = A H
The characteristic separation distance is obtained as eq 8 with the following four assumptions (see Appendix B):
where dimis given by (Harada et al., 1983)
d i m = f,,,,10.4069 + PIm
' 0.9075 In
t
4
Figure 1. Corresponding states correlation for molar volume of alkali halide mixtures. * stands for the component, the mole fractions of which are shown in the figure.
Equations 4a-c and 8 are the mixing rules for the characteristic potential parameters of uniunivalent molten salt mixtures. 3. Corresponding States of Thermodynamic Properties of Alkali Halide Mixtures When the characteristic parameters +, p, 5, and d are determined by eqs 4a-c and 8, respectively, the reduced configuration Helmholtz energy of the mixture is equal to that of the soft-sphere reference and is also equal to that of the hard-sphere reference. Molar volume, V , vapor pressure, p , surface tension, u, and isothermal compressibility, xT, of the molten salt mixtures along the saturation curve can be related to the following universal functions if the density dependency of the d value is neglected: V/(N*d3) P = P ( F ) (10) f
(k)+
6.042
X
lo-(
p d 4 / ( 5 e 2 ) j3 = j3(?)
k)]
ud3/(4e*) = 8 =
(9)
Assumption 4. Fij is specific to the pure molten salt consisting of cation i and anion j and 5 is determined from eq 4b. The terms of-[ij - [ in the first perturbed diagrams in the expansion of A can be neglected, because the values of the Coulomb potential parameters tij (Harada et al., 1983) and [ are about unity. Assumption 5. The exponential of core repulsive potential between the like-charged ions of salts l and 2 is expressed by the arithmetic mean between the unlikecharged ions of salts 1 and 2 (eq B14 in Appendix B). Assumption 6. The average radial distribution function, glm = gCiAm + k C I C m + gAiAm)/2 ( 1 = 1, 2; m = 1, 2), is constant near r = dim,which is the characteristic separation distance for the pure salt consisting of cation l and anion m (Harada et al., 1983), and the gl,'s near r = dlm are equal to each other irrespective of the combination of 1 and m (see eq B17 in Appendix B). Assumption 7. The sum of the integrals over d $0 dlm in the first perturbed diagram in the expansion of A contributes little to those diagrams.
XT[e2/d4
E
2T =
(11)
*(F)
(12)
2T(F)
(13)
where
F = kTd/([e2)
(14)
P and 8 are calculated from the data of 47 mixtures and 26 mixtures, respectively, which are the recommended values of Janz et al. (1974, 1975, 1977, 1979). Figures 1 and 2 show the corresponding states correlations for the molar volume and the surface tension along the saturation curve, respectively, which are evaluated with the help of eqs 10 and 12 with eqs 4b, 8, and 9. The values of J/ij and pij are taken from the report of Tosi and Fumi (1964),and those of Ti.and tijare given in the report of Harada et al. (1983). These values of the four parameters are given in Table I. Mixture LiF-KF reduces to pure LiF and KF at IC = 1.0 and 0.0, and mixture NaF-KF reduces to pure KF at x = 0.0. The data of these pure salts do not fall on the correlation line, while the data of the pure LiF from systems LiF-LiC1, LiF-LiBr, LiF-LiI and LiF-NaF and those of the pure KF from mixtures KF-NaCl, KF-KCl,
1512 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 0
1020 C K 1
c a l c u l a t ed without k. 'I calculated w i t h ki,
Y
>
X.08.02
I
observed
0 7-
-0
0 2.01
X=06,04
( b ) KCIr-l
NaI
1070 C K 1
0
E L .
z 00---------
0
4
0
1
16
17
21
22
oou
23
0
Table I. Interionic Potential Parameters and € i j , ti,, and kij salts hi,W0erg m, lo+ cm 6;; 5;i hi 1.023 0.827 0.970 2.99 2.67 LiF 0.998 0.801 0.988 3.42 3.78 LiCl 0.997 0.845 1.005 3.53 4.41 LiBr 1.013 0.858 0.982 4.30 1.90 LiI 1.000 0.867 0.929 3.30 4.17 NaF 1.OOO 0.947 0.974 3.17 20.1 NaCl 1.002 0.948 0.972 3.40 16.4 NaBr 1.OOO 0.959 0.967 3.86 9.79 NaI 1.004 0.893 0.931 3.38 8.39 KF 1.OOO 1.010 0.963 3.37 28.6 KC1 1.OOO 1.014 0.967 3.35 44.7 KBr 1.006 1.024 0.968 3.55 44.8 KI 1.000 0.905 0.947 3.28 15.5 RbF 1.OOO 0.982 0.959 3.18 72.6 RbCl LOO0 0.989 0.956 3.35 64.7 RbBr 1.004 1.038 0.967 3.37 108 RbI 1.000 0.867 0.960 2.82 98.5 CsF
KF-KBr, and KF-KI just fall on the line. This means that the uncertainty of the experimental data of LiF-KF and NaF-KF is larger than that of the others. V and u can be correlated by eqs 15 and 16 with 1.9 and 6.4% rootmean-square deviations, respectively.
Q = 55.4p -+ 0.384
(15)
b = 0.0289 exp(-46.1?)
(16)
The vapor pressure and the isothermal compressibility were not correlated in the corresponding states, because the data for them were not available. Let the mixing volume, AV, and mixing surface tension, ha, be defined respectively by 2
is1
AU =
(17)
2
u-
CX~U~
i=l
(18)
where V and u are the molar volume and surface tension for the mixtures, respectively, and Vi and ui for pure salt
0.5
10
XNal
Figure 2. Corresponding states correlation for surface tension of alkali halide mixtures. The key is the same as in Figure 1.
AV = V - CxiVi
4'J
IJ
I
1
20
18
l\o/
I " " l " " !
0
1
O O
-
0 0
50 5
w KCI
I
0
-10;
' ' ' ' 0 '5 ' ' '
'
10
LlI
Figure 3. Mixing volume of alkali halide mixtures.
i. Substitution of eq 15 into eq 17 with the help of eqs 10 and 14 yields r
In a similar way, substitution of eq 16 into eq 18 with eqs 12 and 14 yields \
Figure 3 shows comparisons of AV,,,, eq 19, with the observed mixing volume AVO,, where AVdc is shown by a solid line. Examples of the mixtures where A V d , and AVO, agree with each other well are shown in Figure 3a-c. AV,,, has the opposite sign to AVO,, in 6 mixtures that have lithium as a common cation (LiF-LiC1, LiF-LiBr, LiF-LiI, LiC1-LiBr, LiCl-LiI, and LiBr-LiI) and another 12 mixtures (LiF-NaF, NaF-KF, LiCl-KBr, NaC1-NaBr, NaCl-RbC1, KC1-KI, RbC1-RbBr, RbCl-RbI, NaBr-NaI, NaBr-RbBr, KBr-KI, and KBr-RbBr). Figure 3d,e shows examples of the mixtures in which AVdc has the opposite sign to AVO,,. The mixture LiF-LiI has the largest difference between AV,,, and AVO,,. Figure 4 shows comparisons of Au,,,, eq 20, with Auobs, where Aucdc is shown
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1513
-5
( a ) NaBr-KBr
I
1080 [Kl
a,o
oO---o-o-o-
-5 I
i-51
o
I
I
The sign of AgCdc with kij was the same as that without kij in all the mixtures studied. The AaCdc value was not largely affected by kij in all the mixtures, except for the mixtures that incIude LiF or LiI. AV,, cannot be improved with Eij corrected in a similar way to p i j . In the case of + i , , about 20% correction was needed to improve AV,, of tke mixtures that include LiF. Although Cij affects A V, as well as pij, we did not choose fjj as an adjustable parameter because li, itself is a correction factor for the size difference between an anion and a cation of a pure salt. The mixing surface tension was predicted well by using the characteristic parameters 5 and d. The mixing volume was also predicted well by using d and 5 with a small correction factor, kip These indicate that the mixing rules, eqs 4a-c and 8, are reasonable. The mixing rules are used in the corresponding state correlations of transport properties of the alkali halide mixtures in the following paper in this issue (part 2).
observed
- calculated without k
II
c a IcuIa ted with k..
-1 0
m
I
(C)
LiF-LiBr
( b ) NaCI-KBr
1140 C K 1
-- 5 -I0 -,51-
t
1.0 O 5O -u
0.5 KBr
1.0
0.5 LiBr
Figure 4. Mixing surface tension of alkali halide mixtures.
by a solid line. Aaobs can be reproduced by Aucalcwell in almost all the systems studied, as shown in Figure 4. In Figure 4c is shown the mixture LiF-LiBr, which has the largest difference between Aadc and Auob,. A major reason for the opposite sign of AV,, to that of AVobs in 18 mixtures is that the 4 parameters pi,, +ij, Eij, and Sij for the pure salts were used in the mixed salts. An interaction parameter between components of a mixture is not necessary to be equal to that of a pure salt. The effective way to get reasonable AVdc is that the parameter that has the largest effects on the mixing properties among the four parameters is made to be adjustable. In this work, the core repulsive potential parameter pij is corrected with kji as shown by (21) p’ij = [l + (kji - 1)(1- a i j ) ] p i j , i, j = 1, 2 where aij is Kronecker’sdelta function and kji a correction factor. Equation 21 describes that a cross-interaction between a cation in salt i and an anion in salt j is affected by the other cross between a cation in salt j and an anion in salt i when salts i and j are mixed and that the interaction between a cation and an anion in salt i and that in salt j is not affected by each other; that is, these interactions are equal to those in pure salts. By use of p’ij of eq 21 for p i j in eq 9, the sign of AVdc agreed with that of AVoh for all the mixtures studied except for two mixtures, NaC1-RbC1 and KBr-RbBr. AV,,, with p’.. is shown by a broken line in Figure 3. For the mixture gCl-RbBr (Figure 3c), AV, does not change because kij = 1. The AVcal, value agrees with the AVob value satisfactorily. The values of kij are also listed in Table I. The magnitude of correction, Ikij - 11, is about 10-2-2 X for LiF and LiI and less than for the other salts. AVd, values with kij were able to be improved by a small magnitude of correction. The reason why the correction for the salts including lithium ion is larger than that for the other salts may be the larger polarizability of the anions in the lithium halide mixtures than in the other alkali halide mixtures. Addc with k!, is shown by a broken line in Figure 4. Audc for the mixture NaBr-KBr does not change because kij = 1 (Figure 4a) and that for NaCl-KBr does not appreciably change due to kij being close to unity (Figure 4b).
4. Conclusion Mixing rules for the characteristic potential parameters of molten alkali halide mixtures were obtained. By use of the characteristic parameter for the effective Coulomb potential and the characteristic ion distance, the mixing surface tension was predicted well and the mixing volume was also predicted well with a small correction factor, kij.
Acknowledgment Y. Tada gratefully acknowledges the financial support from a grant-in-aid for fundamental scientific research, Ministry of Education, Science and. Culture, Japan (62750854). Nomenclature A = configuration Helmholtz free energy Aij = J . i j / J . - 1 = [ij/[ - 1 c, = p / p i j - 1 D , = coupling parameter of non-Coulombic potential d = characteristic separation distance of mixture dij = characteristic separation distance of pure salt e = elementary charge f = f bond g = radial distribution function = value of radial distribution function in the vicinity of its Bij
maximum k = Boltzmann constant kij = correction factor of a pair between cation i and anion j
N = number of total ions Ni = number of cation or anion of salt i NA= Avogadro’s number p = vapor pressure r = ionic distance T = temperature V = molar volume Xiljm= parameter between the ith ion in salt 1 and the jth ion in salt m (1, m = 1, 2) x l = mole fraction of salt 1 ZN= configuration integral defined by eq A5 zi = valence of the ith ion Greek Letters a = perturbation parameter P = l/(kn y = perturbation parameter A = mixing quantity fij = characteristic parameter of pure salt A = $ exp(-d/p)
1514 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
5 = characteristic potential parameter of the mixture
Fij
= parameter of effective Coulomb potential of pure salt p = characteristic potential parameter of the mixture p i j = parameter of soft-core potential of pure salt u = surface tension @ = total potential defined by eq A6 +ij = pair potential between i and j ions xT = isothermal compressibility IC. = characteristic potential parameter of the mixture Gij = parameter of soft-core potential of pure salt
Superscripts b = blip function C = repulsive part of pair potential H = hard-sphere molten salt HC = hard-sphere potential NC = non-Coulombic potential = reduced form O = reference a y = system with perturbation parameters ' = corrected parameter
(AS)
where Go =
LY
r$$y
ZC@",,
(A91
'>I
Now let X represent the parameters A , B, and C , which are defined by eqs A2a-c, respectively. From assumption 1 in the text, Xil,m between ion i in salt 1 and ion j in salt m is expressed as follows: (AlOa) x c l c l = XAiAi = XCiAl x i i , 1 = 1, 2
and 7
f
Subscripts calc = calculated value like = like-charged ions obs = observed value unlike = unlike-charged ions 0 = reference
Appendix A The pair potential @(r) =
The subscript 0 in eq A7 means a = y = 0. The second term of the right-hand side (rhs) of eq A7 can be written as follows:
XCICm
=
XAlAm
=
(XClAZ
+ x21)/2, + XC2A1)/2 1 # m and 1, m = 1, 2 (AlOb)
= XI,, 1 # m and 1, m
= 1, 2
(AlOc)
where Cl(Al) represents a cation (an anion) in salt 1 and XI,the parameter between a cation in salt 1 and an anion in salt m. Using eqs AlOa-c, and integrating over 3(N - 2) coordinates give eq A8 rewritten as follows:
is rewritten as
+ (1 + aBij)-eiejt r
(Al)
where goEeand gounlikeare the radial distribution functions between like-charged ions and unlike-charged ions of the soft-sphere reference system, respectively. In a similar way, the third term of the rhs of eq A7 can be expressed as
The configuration integral ZgY is defined as ZgY
Sexp(-@W) d N r
(A51
Ex@;? i>]
(A61
p
By use of the characteristic potential parameters $, E, and p , which are expressed by eqs 4a-c in the text, respectively, CC!,m=l xlxmXlm = 0 (X= A, B, or C ) , and eqs A l l and A12 are reduced to the following equations: (A13a)
The logarithm of the configuration integral is expanded in powers of a and y around that of the soft-sphere reference system:
(A13b) Since the higher order terms of a and y of not less than 2 in eq A7 are neglected from assumption 2 in the text, using eqs A13a and A13b and letting a = y = 1 reduce eq ay(
a2i.!:v) + 0
A7 to eq A14 give
... (A7)
In Z N = In Z O N which is eq 5 in the text.
6414)
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1515
Appendix B The reduced configuration Helmholtz energy of the mixture can be expressed in terms of diagrams (Morita and Hiroike, 1961) as
(B13b) The interactions between cations of salts 1 and 2 and between anions of salts 1 and 2 are described as eq B14 from assumption 5 in the text:
-A
-PA = [ s u m of all simple irreducible diagrams composed of two or more black density circles and f bonds] =
The Mayer f bond is defined by
f = exp(-P$) - 1
(B2)
and it is divided into two parts:
f = fH
where
+fb
(B3)
where the dotted bond represents the f b bond. The characteristic separation distance is chosen such that
glm = gClAm + k C I C m + gAlAm)/2, 1 = 1, 2, m = 1, 2 (B16) fiIAm ( I , m = 1, 2 ) is nonzero only near r = dl,,,, and g,, is not so sensitive to r in the vicinity of r = dl,. Thus, it is reasonable that gl, is assumed to be constant near r = dl, in eq B15, and eq B17 is obtained from assumption 6 in the text, although Lantelme and Turq (1979) showed that the smaller ion gives the higher peak of radial distribution function between unlike-charged ions from molecular dynamics calculations for mixtures of LiBr-KBr: (B17) g11 = g22 = g12 = g21 = Using eqs B11 and B17, eq B15 becomes
(B6) Since the second square-blacketed term in eq B5 is neglected from assumption 3 in the text, substitution of eq B6 into eq B5 yields
The integral over d-m in eq B18 is divided into two parts, the integral over d-dl, and that over dlm-m. The latter is evaluated as -drm3/3 (Harada et al., 1983). Thus, eq B18 is rewritten as
= exp(-@$H)- 1
fH
(B4)
Substitution of eq B3 into eq B1 yields
H
+
+
"'
1
+
[sum of simple irreducible
diagrams involving a t least two f b bonds]
(B5)
(B7)
A i A H
1
which is eq 7 in the text. Equation B6 can be described with the radial distribution function for ions i and j as eq B8 (Harada et al., 1983): Z & S g i j ( r ) exp(Mij)f$ d r = 0
038)
i l
exp(@$ij)fican be rewritten in the following way:
dl, is given by eq 9 in the text. Since the value of d is between the minimum and the maximum of the four dl, values (1, m = 1,2), at least one of the four integrals over d-dr, in eq B19 has a contrast sign to the other integrals. Thus, assumption 7 in the text is reasonable, and the sum of the four integrals over d-dim is neglected, which reduces eq B19 to 2
d3 =
where the superscript C means the repulsive part of the pair potential:
4$ = qij exp(-r/pij) $HC=
m,
= $'AIAI = $'CW
$'h
Andersen, H. C.; Weeks, J. D.; Chandler, D. Relationship between the Hard-Sphere Fluid and Fluids with Realistic Repulsive Forces. Phys. Reu. 1971,A4, 1597-1607. Harada, M.; Tanigaki, M.; Tada, Y. Law of Corresponding States of Uni-univalent Molten Salts. Ind. Eng. Chem. Fundam. 1983,22,
(B11)
1 = 1, 2
Pcicr = PAiAi = PCIA~ Pir, 1 = 1, 2 Thus, the following equations hold: &IC1
= 4E1Al = &1Al
which is eq 8 in the text.
(Blob)
From eq AlOa in Appendix A, $'cicl
(B20)
Literature Cited
Equation B9 is reduced to eq B11 from assumption 4 in the text: exP(P4ij)fB = 1 - explP(48 - @HC)I
f,m=l
(BlOa)
r5d
=O, r > d
xx xfxmdfm3
(B12b) (B13a)
116- 121.
Jam, 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, 1-115. 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 1975,4,871-1178. 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:
I n d . Eng. Chem. Res. 1990, 29, 1516-1525
1516
Iodides and Mixtures. Electrical Conductance, Density, Viscosity, and Surface Tension Data. J. Phys. Chem. Ref. Data 1977, 4, 409-596. 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, 125-302. Lantelme, F.; Turg, P. Structure and Diffusion in Mixtures of Ionic Liquids. Mol. Phys. 1979,38, 1003-1014. Lucks, K. D.; Davis, T. Recent Statistical Mechanical Theories of the Thermodynamic Properties of Molten Salts. Ind. Eng. Chem. Fundam. 1967, 6, 194-208.
Morita, T.; Hiroike, K. A New Approach to the Theory of Classical Fluids. 111. General Treatment of Classical Systems. Progr. Theor. Phys. 1961,25,537-578. Reiss, H.; Mayer, S. W.; Katz, J. Law of Corresponding States for Fused Salts. J . Chem. Phys. 1961, 35, 820-826. Tosi, M. P.; Fumi, F. G. Ionic Sizes and Born Repulsive Parameters in the NaC1-type Alkali Halides-11. The Generalized HugginsMayer Form. J . Phys. Chem. Solids 1964,25, 45-52.
Received for review March 6 , 1989 Revised manuscript received January 2, 1990 Accepted January 25, 1990
Law of Corresponding States of Uniunivalent Molten Salt Mixtures. 2. Transport Properties Yutaka Tada,* Setsuro Hiraoka, and Tomokazu Uemura Department of Applied Chemistry, Nagoya Institute of Technology, Nagoya 466, Japan
Makoto Harada Institute of Atomic Energy, Kyoto University, Uji, Kyoto 611, Japan
A law of corresponding states was developed for the transport properties of uniunivalent molten salt mixtures with the use of four characteristic potential parameters and a characteristic mass. the characteristic potential parameters and the characteristic mass were expressed as mixing rules of the potential parameters of the component pure salts and that of the mass of the component ions, respectively. The corresponding states correlation was obtained by expanding the autocorrelation function of the dynamical quantity for the transport property with the potential parameter differences among the component salts and the mass difference among the component ions. The correlation was applied to the electric conductivity and viscosity. The mixing electric conductivity and viscosity were also evaluated, and the mass difference and the difference of softness of the core repulsion largely contributed to the mixing properties. 1. Introduction The law of corresponding states is a useful method for predicting the transport properties of molten salt mixtures, and the estimation of the mixing quantities is the most important in predicting the properties. Young and 0’Connell(l971) proposed a correspondingstates correlation of the transport properties of pure molten salts and their mixtures. The characteristic parameters they used, however, were determined empirically, and the mixing quantities were not evaluated. Harada et al. (1983) simplified the pair potential of molten salt to soft-sphere and effective Coulomb interactions. The effective Coulomb interaction incorporated dispersion and ion polarizability effects. The thermodynamic properties of the pure molten salts were correlated in correspondingstates with the parameter of the effective Coulomb potential and the characteristic ionic distance determined by scaling the soft-sphere potential to a hard-sphere potential. Tada et al. (1988) showed that the transport properties of the pure molten salts were correlated in corresponding states with the simplified potential parameters, the characteristic distance determined by Harada et al. (1983) and the characteristic mass obtained by expanding the autocorrelation function of the dynamical quantity for the transport property with the mass difference of anion and cation. The aim of this paper is to investigate what parameters largely contribute to the mixing transport properties of uniunivalent molten salts on the basis of the simplified pair potential of Harada et al. (1983). The transport property, which is expressed in terms of the time correlation of the dynamical quantity for the 0888-5885/90/2629-1516$02.50/0
property, is reduced by the characteristic mass and the characteristic potential parameters and ionic distance and is expanded with the mass difference of the component ions and the differences of the potential parameters. The characteristic mass is chosen such that the first perturbed term with respect to the mass difference vanishes and the characteristic potential parameters and ionic distance are chosen such that the Helmholtz energy of the mixture equals that of the reference salt as shown in part 1 (preceding paper in this issue). The transport property of the mixture is given by the sum of the transport property for the reference system of anion and cation with the unique characteristic mass in which the ions interact through the simplified potential with the characteristic potential parameters and of the perturbation terms with respect to the mass difference and with respect to the cross term of the mass difference and the difference of the softness of the core potential. By use of this perturbation theory, the electric conductivity and viscosity of the mixture are obtained. The mixing electric conductivity and viscosity are evaluated by using the equations for the corresponding states correlations for the pure salts and the mixtures, and it is shown that the cross term of the mass difference and the difference of softness of the core repulsion largely contributes to the mixing properties. 2. Perturbation of Transport Properties
The pair potential between ions i and j of a uniunivalent molten salt mixture is expressed by +ij(r)
= $i; exp(-r/pi;)
0 1990 American Chemical Society
+ eiejEi;/r
(1)
