Ind. Eng. Chem. Res. 1988,27, 1042-1049
1042
Corresponding States Correlation of Transport Properties of Uniunivalent Molten Salts Yutaka Tada,*t Setsuro Hiraoka,+Tomokazu Uemura,+and Makoto Haradat Department of Applied Chemistry, Nagoya Institute of Technology, Nagoya 466, Japan, and Institute of Atomic Energy, Kyoto University, Uji, Kyoto 611, Japan
A law of corresponding states was developed for the transport properties of uniunivalent molten salts with the use of four potential parameters and a characteristic mass. The characteristic mass was defined as mR = 2mAmc/(mA+ mc) or ms = {2mA1/2mC1/2/(mA1/2 + mC1/2))2, where mA and mc are anion and cation masses, respectively. The corresponding states correlations were obtained by expanding the autocorrelation functions of the dynamical quantities for the transport properties with the mass difference between the anion and cation. The four-potential-parameter correlation was applied to the electrical conductivity, viscosity, and self-diffusion coefficient. The four-parameter correlation was reduced to a two-parameter correlation with the use of the characteristic molar volume and temperature which were specific to each salt. mC1/2))2, which was proposed by Ebbsjo et al. (1974) and Lanthelme et al. (1976, 1977). The transport properties for the molten salt, which are composed of the anion and the cation with different masses, are given by the sum of the transport properties for the reference system of the anion and the cation with a unique characteristic mass, and of the perturbation terms with respect to the mass difference. It is shown that satisfactory correlations for the transport properties of the uniunivalent molten salts can be obtained with the help of the corresponding states equations described by using the perturbation theory with respect to the mass difference.
1. Introduction The law of corresponding states is useful for predicting the transport properties of molten salts. Young and O'Connell (1971) proposed an empirical corresponding states correlation of the transport properties of uniunivalents molten salts. There is, however, no theoretical basis for the characteristic parameters involved in their correlation. White and Davis (1967) showed a corresponding states correlation for the thermal conductivity, A, of molten alkali nitrates by using the following assumptions: (1)Pair potential between the ions is comprised of the short-range repulsive and the Coulombic potentials, and only the short-range repulsive potential between the unlike ions is important among the short-range potentials. (2) The collisional energy transport is accomplished predominantly by the nearest-neighbor collisions. The first assumption, which is the same as postulated by Reiss et al. (1961), is ascertained to be reasonable by referring to the ionic configuration. From the second assumption, twice the reduced mass between the unlike ions would be chosen as the characteristic mass for the correlation. The transport properties, K , are expressed in terms of the time correlation of the dynamical quantity A by the fluctuation-dissipation theorem:
2. Perturbation of Transport Properties with Respect to Mass Difference The size difference between unlike ions plays a minor role in the transport properties, if the difference is not large. I t is, therefore, postulated that the pair potential between i and j ions is expressed by a function of the separation distance r:
dij(r)= )I exp(-r/p)
where t = time, p is l / k T , and V is the volume of a system. The time correlation ( A ( O ) A ( t ) is ) generally affected not only by dynamical processes in short times but also by collective mode of ionic motion in long times. The second assumption made by White and Davis (1967) is, in general, not valid for the corresponding states correlation of transport properties of molten salts. Thus, the reduced mass does not always represent the characteristic masses of these systems. The aim of this work is to describe the corresponding state equations of transport properties of uniunivalent molten salts, by using a perturbation theory with the mass differences between the anion and cation as the variable. Two kinds of effective masses are used for the characteristic mass: One is twice the reduced mass between the unlike ions, mR = 2mAmc/(mA+ mc),where mAand mc are the masses of the anion and cation, respectively. An+ other is the mass defined by ms = {2mA11zmc1~z/(mA1~z
+ zizje2t/r
(2)
where 4 and p are the potential parameters of repulsion and are taken to be the ones for the unlike ion pairs. The second term in eq 2 represents the Coulombic potential between i and j ions with the valences zi and zj, respectively. t represents a parameter specific to the salt species, which incorporates the effects of the weak long-range potential and the dielectric constant. The pair potential, eq 2, is rewritten in a reduced form: & j ( i= ) d i j ( i ) / A = exp(-d(i - l ) / p )
A =
+ zizje2t/(Adi)
IFi exp(-d/p)
(3)
(4)
(5) where d represents a characteristic separation distance between the unlike ion pair. The momentum of the ith ion p i is reduced to a nondimensional form: i = r/d
pi = pi/(AAi)'l2
(6)
where i+ii is an arbitrary mass. The Hamiltonian is expressed in a reduced form: H = H / A = Ho + &
Nagoya Institute of Technology. Kyoto University.
(7) 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1043 Here,
& = @/A,
@ =
C&j,
Pi =
i>j
(mi/%) - 1 (8)
The product of the Liouville operator L and time t is written as itL = iiL = iiLo i i d
+
iLo =
C(Pia/aFi - a&/aPi a/aii)
perturbed terms of the Hamiltonian, the evolution operator, and the dynamical quantity A ( 0 (Ebbsjo et al., 1974): (A(o)A(t))fi = exp(iiLo + i i h i ) i ( o ) )Go+&
(A(o)
(A(o)A(t^))
=
(A(o) exp(iiL,)A(O)) 8, + 5 ~ R , ~ (i i ) ~ k i n n=l
(18) Introducing eq 18 in eq 11 yields
1
m
KR i
t^ = ( t / ~ l ) ( A / f i ) l / ~ P/ = ( f i / f i i ) 1 / 2
-1
(10)
where f i is a characteristic mass for a given molten salt. The subscr_ipt o in-eq 7 and 9 represents the reference system. AH and hL are the perturbed terms with respect to the mass difference between the anion and cation. The collective transport properties, i.e., electrical conductivity, viscosity, and thermal conductivity, can be expressed with the help of fluctuation-dissipation theorem: 1 K = ( A ( t ) A ( O ) ) , dt VkT o
-1
where A ( t ) is the dynamical quantity and A is its time derivative. The brackets ( ,) represent the canonical average with respect to the Hamiltonian, H. The transport properties conjugated to A ( t ) ) are shown in Table I, with the reduced form of A ( i ) .,A(i) is divided into the unperturbed reference term A0(0 and the perturbed term AA(8. These terms are also shown in Table I. Equation 11 is expressed in a reduced form by
I? = ( K V k T d / A 2 ) ( f i / A ) 1=/ 2 Jm(A(i)A(0))fi dt^ (12) The masses, f i i and f i , which are introduced in eq 6 and 9, are properly chosen. We choose these masses in two ways: f i i = f i = mR = 2mAmc/(mA+ mc) (13) (1) (11)
=
KRo
m
+ CSR,nCpi" = R R o + i
n=l
sR,2mkR2m m=l
(19)
where KRois the reduced transport property of the reference system, which is composed of the anion and the cation with unique mass mR: KRo
= Jm(Ao(f)Ao(0)) f i 0 d i
The odd terms of pin ( n = odd) in eq 18 vanish because of eq 17. The second choice, eq 14, is based on the fact that the perturbed Hamiltonian vanishes (Ebbsjo et al., 1974),
A = Ho
(21)
and the perturbed term of the evolution operator is reduced to ihi = pi
Eoi a/aFi - a&/aFi a/api)p; i
+
= ps = (mA1/2- mC1J2)/(mA2 mC112) i pi'
= ps
i E anion
(22)
E cation (23)
In this choice, the mass difference is characterized only by the term ps, and the perturbed term for viscosity vanishes (Table I). The autocorrelation function is expressed by the sum of the term for the reference system and the perturbed terms arising from the perturbation terms in evolution operator and A(f): (A(o)A(t^))fi = (A(o) exp(iiLo + iiAL)A(o))fi
fii = mi,
+
f i = ms = (2mA1/2mc1/2/(mA1/2mC1/2)]2 (14)
The first choice, eq 13, is based on the fact that the autocorrelation function is well expressed by the relative motion of pair particles on the short time scale. The Hamiltonian is expressed in a reduced form:
H = Ho + C B i 2 4 2
(15)
i
The evolution operator is i d = where
hi
iLo + Coi a/ari)pi i
(16)
(mA- mc)/(mA Pi
= -PR
+ mc)
i E anion
I?, is the transport property for the reference system with the mass ms. The odd terms with respect to n in eq 24 vanish due to eq 23. Self-diffusion coefficient is a transport property of singlet motion and is related to the friction coefficient li: 5;. = kT/(Dimi) i E cation, anion (26) Harada et al. (1982) showed that the perturbation with respect to the mass difference for the friction coefficient provided
is expressed as
pi = pR =
The resultant transport property K s is expressed by
i E cation (17)
In this choice, the mass difference is characterized only by the parameter pCLR. The molten salt considered in this work is a symmetric salt composed of uniunivalent cation and anion which interact through the same repulsive potential. Taking this into account, the autocorrelation function is expressed by the s u m of the reference term and the perturbation terms with respect to the mass difference pR arising from the
m
l / f i i ~=
(kTd/Di)(mRA)-"' =
l/fi;~ + ~Z s ~ , d ~ & ~ n=l
(27)
l/fiis = (IzTd/Di)(msA)-1/2= 1/Bis0+
m
CSs,dnpi'" n=l
(28)
Since the self-diffusion coefficient is the transport property of singlet motion, the odd terms of K~~ and p i n do not
1044 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 Table I. Dynamical Quantities and Their Perturbed Forms transport properties, K electrical conductivities,
'40 = A,(O + osi(f)
A [i(f) = .4(t)/Ai
A(t)
ii.(t)
X 2 P Z 1 L
I
viscosity, q
A, = mR, A = mR
transport properties
K*
A, = m,,f i = ms
electrical conductivity
viscosity
0
thermal conductivity
( V / N A d 3 )f,
=k T d l ( e 2 [ )
vanish. This is in contrast to the nature of the collective transport properties. 3. Law of Corresponding States of Transport Properties In the preceding section, we obtained the perturbation formulas for the transport properties, eq 19, 25, 27, and 28. These equations can be applied to the systems which interact through the interionic potential of conformal type. The interionic potential in the form eq 3 has two potential parameters, e z , $ / ( A d )and d l p , which take different values in a given class of molten salts. It is important how the reduced forms of the transport properties should be expressed for the reference systems with the parameter sets e 2 E / ( A d )and d l p . Rice (1962) showed that the Coulombic potential would play a minor role in the transport properties, because the Coulombic force is smeared out around an ion due to the long-range nature of the Coulombic potential. Thus, the scaled Coulombic potential parameter e 2 t / ( A d )has a minor effect on the transport properties. Taking this into account, the reduced form of the collective trarpport properties, K , can be replaced by the simple form K * as shown in Table Il by discarding the terms V, T , and e 2 E / ( A d )in the term K. The perturbation formulas, eq 19 and 25, are rewritten as P
= K*Ro
+ m=l C S*R,2,,,pRZm
(19')
m
K*, = K * s o +
c S*S,2mps2m
m=l
(25' )
Generally, K * and K*Dare given as functi_onsof the reduced state variables, T (=kTd/(e2,$))and V (= VI (NAd3) ( N A= Avogadro's number), and of the softness parameter of the core repulsive potential, d l p . The characteristic size of the unlike ion pair, d , is selected such that the perturbation term with respect to softness of the repulsive core potential vanishes in the perturbation expansion of the excess Helmholtz energy; that is, the Helmholtz energy is equivalent to that in the
system of the hard core repulsive and the Coulombic potentials. In other words, d is selected to be equal to the minimum separation distance d* between the unlike ions in the latter system. When this minimum separation distance, d*, was used as the characteristic size of the unlike ion pair, d , it was reported by Harada et al. (1983) that the thermodynamic properties of alkali halides were well discribed by the law of corresponding states. The V along the saturat9d liquids line was expressed by a universal function of T for alkali halides, if the d* was selected as d. Thus, the reduced form: of the cpllective transport properties in eq 19' and 25', K * and K o, along the saturated liquids line would mainly depend on the T. The self-diffusion coefficients of eq 27 and 28 can be rewritten by the same reasoning mentioned above: m
1 / 8 * p , = (d/Di)(h/mR)'i' = l/B*i~,,
+ cs*d~,~pj~ n=l
(27'
l/B*is = (d/Dj)(A/mS)'/' = 1/B*s0 +
m
n=l
,!?*dS,npiln
(28' 4. Correlations of Transport Properties for Alkali Halides The values of the parameters, p and $, for alkali halides are taken from the report of Tosi and Fumi (1964). The values of d* are calculated from the following equation proposed by Harada et al. (1983): d * / p = ([0.4069 + 0.9075 In ( $ / k T ) 6.042 x lO-'$/kT] (29)
+
where the values of t and { given in the report of Harada et al. (1983) are shown with the values of p and $ in Table 11. 4.1. Electrical Conductivity. Figure l a shows the temperature dependence of the reduced electrical conductivities, ii*R, of molten alkali halides, where the characteristic mass is given by eq 13. As the plots of the reduced electrical conductivity against the pR2 at any reduced
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1045 Table 11. Interionic Potential Parameters and € and salts +, erg p , IO4 cm € c 0.827 0.970 2.99 2.67 LiF 0.801 0.988 3.42 3.78 LiCl 0.845 1.005 3.53 4.41 LiBr 0.858 0.982 4.30 1.90 LiI 4.17 3.30 0.867 0.929 NaF NaCl 20.1 3.17 0.947 0.974 16.4 3.40 0.948 0.972 NaBr 9.79 3.86 0.959 0.967 NaI 8.39 3.38 0.893 0.931 KF 1.010 0.963 3.37 28.6 KCl 1.014 0.967 3.35 44.7 KBr 1.024 0.968 3.55 44.8 KI 0.905 0.947 3.28 15.5 RbF 0.982 0.959 3.18 72.6 RbCl 0.989 0.956 3.35 64.7 RbBr 1.038 0.967 3.37 108.0 RbI 98.5 2.82 0.867 0.960 CsF
I C.
F CI Br I L I O O O *
:r"
~
I
I
1
2
,
I
,
I
,
3 4 5 678910 K, (calculated) [ S.cm-'l
Figure 2. Comparison of the electrical conductivities calculated with the observed ones, K,,&. Key is the same as from eq 30, in Figure 1.
0.08 N
,
m O m e
"."I
0.04
I
0.03 0.02
1
i 50
40
60
l l i
-
Figure 1. Correlation of the electrical conductivitiesof molten-alkali halides with the use of characteristic mass mR. (a) R*R vs 1/T. (b) R * b vs 1/T.
temperatures give parabolic curves, the reduced electrical conductivity is expressed from eq 19' as ?*R
The
8*eR2
and
s*eR2
= ?*Ro s*&4
=0
+ s * e R $ l R 2 + $*&4FR4
(30)
are expressed as follows: s*eR4
= -7.4 x
(31)
Figure l b shows the reduced electrical conductivities of the reference-system, ii*&, which are evaluated from ( k * R - S * e ~ + R 2 - S*eR#R4). The ii*Ro can be correlated by the following equation within 10% deviation regardless of the species of molten alkali halides: ii*Ro = 0.181 exp(-O.O304/p) (32) The coefficients in eq 31 and 32 and also those in eq 34, 35, 37, 38, 40, 41, 43, 44, 46, and 47 stated below are evaluated by the least-squares fitting of the data. Figure 2 shows the comparison of the electrical con~ ~ , the observed ductivities calculated from eq 30, K R , ~ with ones, Kobs, where the root-mean-square deviation of from Kobe is 6.5 % . When the characteristic mass is given by eq 14, the reduced electrical conductivity ii*s is correlated as shown in Figure 3a. The at any reduced temperatures is also
0.071
I
Figure 3. Correlation of the electrical conductivitiesof molten-alkali halides with the use of characteristic mass mS. (a) R*s vs 1/T. (b) R*So vs 1/T. Key is the same as in Figure 1.
expressed by a parabolic curve with respect to similar fashion to eq 30: ii*s = ii*so
+ s*eS2ps2 + S * e S 4 p S
$*eS2
= 3.33P - 0.0545
$*eS4
= -15.9p
+ 0.264
4
ps2 in
a
(33) (34)
The reduced electrical conductivity for the reference system, is shown in Figure 3b. The ?*so can be correlated by the following equation within 11% deviation: ?*so = 0.181 exp(-O.O304/p) (35) The root-mean-square deviation of the calculated K S from the observed ones is 6.7 % . The mass perturbation terms have weak effect on the electri-cal conductivity ii*, so both ii*R and ii*s are correlated with T with similar precision. 4.2. Viscosity. Figure 4a shows the temperature dependence of the redused viscosity, jj*R, for alkali halides. The viscosity at any T i s varied linearly with the pR2value. Then, the reduced viscosity is expressed as il*R = $*Ro + S * v R 2 ~ R 2 (36) The S*vR2 is expressed by a linear function of lip: S'*VR2 = 0.229/? - 8.61 (37) The reduced viscosities for the reference system, jj*Ro,
1046 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 i
1
*
i
00
* %
00 0
It
*i
(b)
*
50
40
50
I/
i
60
I N V I
i
43-
f J :
-
I
*h. 3
2-
C *Fm
50
40
I/?
60
which are evaiuated from (+j*!- f i * v R 2 p ~ 2 ) , can be correlated with 1 / T as shown in Figure 4b, where ij*Ro is expressed by +j*Ro = 0.0772 exp(0.0621/ f') (38) The viscosity of alkali halides can be predicted from eq 36-38 within 17% deviation, excluding lithium chloride, lithium iodide, and cesium fluoride, which show large mass differences between the anions and cations. The rootmean-square deviation of the T R , from ~ ~ qobs is 11% for all molten salts shown in Figure 4. The reduced Viscosity was plotted against ? in Figure 5a. From this, the reduced viscosity for the reference system, is evaluated as shown in Figure 5b, with the help of
+ s*vszcLs2 Ljl*vs2 = 0.366/f' - 13.1 ?*s =
?*so
l l i
60
Figure 5. Correlation of the viscosities of molten-alkali halides with the use of characteristic mass m g (a) fj*s vs l/T. (b) ?*so vs 1/T. Key is the same as in Figure 1.
Figure 4. Correlation of the viscosities of moltenAalkalihalides with the use of characteristic mass mR. (a) f j * ~vs 1/T. (b) fj*bvs 1/T. Key is the same as in Figure 1.
The
LO
(39) (40)
is expressed by
= 0.0608 exp(0.0683/?) (41) The viscosity of alkali halides can be evaluated by eq 39-41 within 12% deviation, excluding lithium chloride, lithium bromide, lithium iodide, potassium iodide, and cesium fluoride. Figure 6 shows the comparison of the viscosities calculated from eq 39, qs,cdc,with the observed ones, Ifobs. The root-mean-square deviation of T S , from ~ ~ ?lobs ~ ~ is 9% for all molten salts shown in Figure 6. The corresponding states correlation with the characteristic mass, ms, is better than that with mR. When ms is selected as the characteristic mass, one can expect several merits: (1)The Hamiltonian is invariant regardless of mass difference. (2) The momentum current flow has no perturbed term. (3) ps2 takes a smaller value than pR2in general, and the perturbed series with respect to ps2 con-
r
-
l
'
l
I
'
*'
,'
m 2 0 a E
05v'1 05
' ' ' '
1
1 2 rl, (calculated) [ mPa s
,
I
Figure 6. Comparison of the viscosities calculated from eq 39, T with the observed ones, qOh. Key is the same as in Figure 1.
~ , ~ ~ ,
verges more rapidly in ms than in mR. The viscosity is well correlated by the use of ms with the above merits, even when the dynamical process of lower frequencies contributes to the viscosity. 4.3. Self-Diffusion Coefficient. I t was reported by Harada et al. (1982) that the mass effect on the self diffusion coefficient was described by eq 42 with the help of the molecular dynamics calculation: l/BrR
= 1/B*l,
+ s*dR1pL2+ 9*dR2p12
(42)
Figure 7a shows the temperature-dependency of the reduced self-diffusion coefficient, D*jR. Figure 7b is the temperature dependency of the l/D*,Rocalculated from eq 42 by assuming = -13.0,
= 4.77
(43)
The fi*&Ro is expressed by l/l?*IRo = 1.27 exp(0.0634/?)
(44)
b!?*d,2
Figure 8 shows the comparison of the self-diffusion coefficients calculated from eq 42, DfidC,with the observed
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1047 60-
501
40 -
--
LO ( a )
300
20 -
I
601
5O-(a)
0
.
N b /NaC I CI-/NoCI K' i K C l
g2301
n D
0
- 20
Rb*/RbCI CI-/RbCI Na'i Na I I- /Nal
0
IO LO
50
I /
i
60
,.,
40
Figure 7. Correlation of the self-diffusion coefficients of qolten alk-ali halideswith the u,se of characteristic mass mR. (a) l/D*iR vs 1/T. (b) l / D * i ~vs, 1/T.
Figure 8. Comparison of the self-diffusion coefficients calculated from eq 42, l/D,R,edowith the observed ones, l/Di,oh Key is the same as in Figure 7.
ones, Di,obs. The self-diffusion coefficients for sodium chloride and iodide, potassium chloride, and rubidium chloride can be well correlated by eq 42 except for the self-diffusion coefficient of chloride ion in rubidium chloride. The root-mean-squaredeviation of l / D l ~from ~e l/Dj,absof these ions is 7%. The l/Di,obisof the lithium and the chloride ions in lithium chloride, the sodium ion in sodium fluoride, and the potassium ion in potassium fluoride are about 30% lower than those of the other ionic species mentioned above. The root-mean-square deviation of l/DiR,cdcfrom l/Di,obeis 27% for all i p s in Figure 8. The rejuced self-diffusion coefficient D*is was plotted against T i n Figure 9a. The redyced self-diffusion coefficient for the reference system, D*jso,is shown in Figure 9b with the help of 1/8*iS = 1/8*iSo + S * d S l p [ S*dS1
= -20.7,
S*&
+ S*dS2/.b[2 = 11.9
The B*iso is expressed by 1/B*;s0= 1.36 exp(O.O619/T)
(45) (46) (47)
The self-diffusion coefficient is correlated with the use of ms with a similar precision to that with the use of mR.
60
501/i
Figure 9. Correlation of the self-diffusion coefficients of molten alkFli halide5with the y e of characteristic mass ms. (a) 1/D*is vs 1/T. (b) l/D*iso vs 1/T. Key is the same as in Figure 7. Table 111. Characteristic Molar Volume and Temperature salt To, K V,, cm3/mol LiF 1246 14.60 LiCl 987 28.83 LiBr 962 35.15 LiI 889 44.43 NaF 1168 20.74 1048 NaCl 37.09 NaBr 994 43.18 933 NaI 54.05 KF 1084 29.71 KC1 1049 48.91 KBr 969 54.89 KI 909 66.19 1044 RbF 35.52 RbCl 963 52.98 RbBr 911 72.81 CsF 876 39.65 CSCl 919 59.85 CsBr 897 67.11 CSI 928 82.02
We cannot obtain a better corresponding states correlation for the self-diffusion coefficient than those for the electrical conductivity and viscosity. This would mainly arise from the fact that the observed self-diffusion coefficients contain considerable uncertainty. For the thermal conductivity, the corresponding states correlation was not obtained due to lack of the experimental data. We believe, however, that satisfactory correlation could be obtained for the thermal conductivity by using the present approach. This is left to future work.
5. Corresponding States Correlation with the Use of Characteristic Parameters In the preceding section, we proposed the corresponding states correlations of the transport properties with use of the four potential parameters, $, p , 5, and {, which were evaluated with the help of thermodynamic properties. Since the ionic diameter determined from eq 29 is a weak function of the temperature, a constant value in the limited temperature range can be assigned approximately to the diameter for each salt. Also, the softness of the core potential would have minor effect on the transport properties.
1048 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
Figure 10. Comparison of the electrical conductivities calculated ~ ~ the ~ observed , ones, K , ~ . Key is the same as from eq 48, K ~ , with in Figure 1.
2
I T 5 (calculated)
05
L: mPa s 1 Figure 11. Comparison of the viscosities calculated from eq 57, ? ~ , ~with d ~ the , observed ones, )lobs. Key is the same as in Figure 1.
Thus, the four-parameter law of corresponding states described in the preceding section can be reduced to the two-parameter one, in which the two parameters are characteristic molar volume (V,) and temperature (To). With the use of these two parameters, the electrical conductivity, the viscosity, and the self-diffusion coefficient are described as follows:
+
(48)
= 0.129 exp(-1.03/p)
(49)
BR = K / ~ ~ ( V ~ / N A ) ' ' ~ ( = ~T k~~~ ~ S Re)~' 2/ p' ~ ' RRo
SeR2
(50)
=0
xs = K / ~ ~ ( V , / N ~ ) ~ =/ Rso ~ (+~seS2~S2 T , ~ ~ ) ~(51) ~' Rso = 0.137 exp(-l.O7/p)
(52)
SeSP= -0.0114p + 0.0241
(53)
= 24.4
(65)
4
5F
1 / D I', ~
,
I
20 30 40 X10~3(calculated) 1 1 10 C cm-251
Figure 12. Comparison of the self-diffusion coefficients calculated from eq 63, l/DtS,cdc,with the observed ones, l/Dt,oba.Key is the same as in Figure 7.
The correlation of the electrical conductivity with the use of eq 51 was obtained with similar precision. The twof~ = o(vo/N~)2i3/(kTom,)1/2 = + s v ~ 2 F ~ ' (54) parameter correlations, eq 48 and 51, do not contain the p4term, even though the four-parameter correlations, eq ?jRo = 0.552 exp(2.51/p) (55) 30 and 33, do, because the perturbed term has little effect on the electrical conductivity and the relatively large S v ~= 2 13.9/T- 7.86 (56) scattering of the data in the two-parameter correlations fS = ~ ( ~ O / N A ) ~ / ~ / ( ~ = T?jSoO+~ sSv S) 2' ~"S 2 (57) covered the effect of the perturbed term. Figure 11 shows the comparison of the viscosities cal= 0.385 exp(3.02/p) (58) culated from eq 57 with the observed ones. The rootmean-square deviation of qs,calc from qObs is 14%. The SvSz = 19.5/F - 10.5 (59) correlation of the viscosity with the use of eq 54 was obl/b, = ~/D;(V,/NA)''~(~T~/~R)'/~tained with similar precision. Figure 12 shows the comparison of the self-diffusion (60) = l / b i R o + 3&1&+ SdR2P: coefficients calculated from eq 63 with the observed ones. l/bi%= 1.09 exp(3.49/7+) (61) The root-mean-square deviation of l/D,s,cdcfrom l/D)r,obs is 27%. Excluding sodium ion in sodium fluoride, poS d R l = -14.9, SdR, = 8.96 (62) tassium ion in potassium fluoride, lithium and chloride ion in lithium chloride, and chloride ion in rubidium chloride, 1/bjs = 1/Di( v O / N ~ ) l i 3To/ms)1/2 (k the root-mean-square deviation of l/Di,cdcfrom l/DL,obs (63) = l/fiiSo + SdSlk[ + SdSzWl is 7.6% with the use of the characteristic mass (ms)and is 9.0% with the use of mR. (64) l/Diso = 1.09 exp(3.47/T)
-
sds1
= -24.5,
sds,
p = T/To
(66)
where NA is Avogadro's number. The characteristic molar volume and temperature were decided by Harada et al. (1983) 90 that the thermodynamic properties can be well correlated by using these parameters. The parameter values are shown in Table 111. Figure 10 shows the comparison of the electrical conductivities calculated from eq 48 with the observed ones. The root-mean-square deviation of K R , ~ & from Kobs is 13% .
6. Conclusion A law of corresponding states was developed for the electrical conductivity, viscosity, thermal conductivity, and self-diffusion coefficient of uniunivalent molten salts. A four-potential-parameter correlation was applied to the electrical conductivity, viscosity, and self-diffusion coefficient. For the electrical conductivity, both correlations with the use of mRas the characteristic mass and with the use of ms were obtained with similar precision. For the viscosity, the correlation with the use of ms showed a smaller root-mean-squaredeviation of the calculated values
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1049 from the observed ones than that with the use of mR.The four-parameter correlation could be reduced to a two-parameter correlation with the use of the characteristic molar volume and temperature which were specific to each salt. Experimental data except for the viscosities for LiF and NaCl are from the recommendation by Janz (1980),Janz and Bansal (1982), and Janz et al. (1974,1975,1977). The viscosity data for LiF and NaCl are from Abe et al. (1980, 1982). Acknowledgment
M. Harada gratefully acknowledges the financial support from a grant-in-aid for fundamental scientific research, Ministry of Education, Science and Culture, Japan. Nomenclature A = dynamical quantity ?(t) = time derivative of dynamical quantity A(t) A(0 = time derivative of dynamical quantity A(t) in reduced form AA(8 = perturbed term of time derivative of dynamical quantity A(t) in reduced form Di= self-diffusion coefficient of the ith ion Qi = reduced self-diffusion coefficient of the ith ion Di = reduced self-diffusioncoefficientof the ith ion in simple form d = characteristic separation distance d* = hard sphere diameter e = elementary charge = Hamiltonian H=, reduced Hamiltonian defined by eq 7 AH = perturbed term of reduced Hamiltonian defined by eq 7 i& = Liouville operator iL = reduced Liouville operator i d = perturbed term of reduced Liouville operator defined by eq 9 K = transport property K = reduced transport property k = Boltzmann constant mA = anion’s mass mc = cation’s mass mi = mass of the ith ion mR = characteristic mass defined by eq 13 ms = characteristic mass defined by eq 14 f i = characteristic mass Ai= arbitrary mass N A = Avogadro’s number pi = momentum of the ith ion pi = reduced momentum of the ith ion r = interionic distance 4 = reduced distance defined by eq 5 S, = reduced coefficient of the nth term in perturbation expansion of transport property 3, = reduced coefficient of the nth term in perturbation expansion of transport property in simple form s^,(t) = reduced coefficient of nth term in perturbation expansion of autocorrelation function T = temperature To= characteristic temperature T = kTd/(e2t) = reduced temperature t = time f = reduced time defined by eq 9 V = volume yo = characteristic volume v = v/(NAd3)= reduced volume
e
Greek Symbols p = l/kT f = parameter defined by eq 29
ti = friction coefficient
= viscosity .il = reduced viscosity 9 = reduced viscosity in simple form K = electrical conductivity 2 = reduced electrical conductivity = reduced electrical conductivity in simple form A = parameter defined by eq 4 h = thermal conductivity p R = mass difference parameter defined by eq 17 ps = mass difference parameter defined by eq 23 pi = mass difference parameier defined by eq 8 p i = mass difference parameter defined by eq 10 6 = parameter modifying the reduced temperature p = parameter of the soft core potential = total potential of the system defined by eq 8 = reduced total potential of the system defined by eq 8 dL,= pair potential between i and j ions $ = parameter of the soft core potential Q
Subscripts calc = calculated value d = self-diffusion coefficient e = electrical conductivity o = reference system obs = observed value R = reduced form with use of characteristic mass mR S = reduced form with use of characteristic mass ms v = viscosity Superscript * = reduced transport property in simple form Literature Cited Abe, Y.; Kosugiyama, 0.;Nagashima, A. Nihon Kikaigakkai Ronbunshu B 1982, 48, 149 (viscosity for LiF). Abe, Y.; Kosugiyama, 0.; Miyajima, H.; Nagashima, A. J . Chem. Soc., Faraday Trans. 1 1980, 76, 2531 (viscosity for NaCI). Ebbsjo, I.; Schofield, P.; Scold, K.; Walker, I. J . Phys. C. Solid State
Phys. 1974, 7, 3891. Harada, M.; Tanigaki, M.; Tada, Y. Ind. Eng. Chem. Fundam. 1983, 22, 116. Harada, M.; Yamanaka, A.; Tanigaki, M.; Tada, Y. J . Chem. Phys. 1982, 76, 1550.
Janz, G. J. J . Phys. Chem. Ref.Data 1980, 76, 791 (electrical conductivity for NaC1). Janz, G. J.; Bansal, N. P. J . Phys. Chem. Ref.Data 1982, 11, 505 (diffusion coefficient). Janz, G. J.; Gardner, G. L.; Krebs, U.;Tomkins, R. P. T. J . Phys. Chem. Ref.Data 1974,3, 1 (electrical conductivity and viscosity for fluorides). Janz, G. J.; Tomkins, R. P. T.; Allen, C. B.; Downey, J. R., Jr.; Gardner, G. L.; Krebs, U.; Singer, S. K. J . Phys. Chem. Ref.Data 1975, 4,871 (electrical conductivity and viscosity for chlorides). Janz, G. J.; Tomkins, R. P. T.; Allen, C. B.; Domey, J. R., Jr.; Singer, S. K. J . Phys. Chem. Ref.Data 1977, 4,409 (electrical conductivity and viscosity for bromides and iodides). Lanthelme, F.; Turq, P.; Schofield, P. Mol. Phys. 1976, 31, 1085. Lanthelme, F.; Turq, P.; Schofield, P. J . Chem. Phys. 1977,67,3869. Reiss, H.; Mayer, S. W.; Katz, J. J . Chem. Phys. 1961, 35, 820. Rice, S. A. Trans. Faraday Soc. 1962,58, 499. Tosi, M. P.; Fumi, F. G. J . Phys. Chem. Solids 1964,25, 45. White, L. R.; Davis, H. T. J . Chem. Phys. 1967,47, 5433. Young, R. E.; O’Connell, J. P. Ind. Eng. Chem. Fundam. 1971, 10, 418.
Received for review September 23, 1987 Accepted January 4, 1988
