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)
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1517
6 s ( V / N A d S )T, = k T d / ( e a € ) A ( t ) / A = a(0 = Ao(i) + AA(0
Table I. Dynamical Quantities and Their Perturbed Forms.
transport properties electrical conductivities
K
A(t) &erh
K
A
Crm,
K*
A
The reduced form of the pair potential is divided into two parts, the reference and the perturbed terms:
$"ij(i)
K
i
viscosity
ai,(?)
AA(0
A O ( 0
+
= dij(r)/A = aoij(i) AJij(i) = exp{-d(i - l ) / p J
+ eiej[/(Adi)
(2) (3)
The product of the Liouville operator iL and time t, is also divided into the terms for the reference system iLof and the perturbed terms:
iLt = iLt^ = iLOt^+ idmi+ idQt^ + idmQt^(13)
A&,@) = exp(-d(i - l ) / p J ( ( l + Ai,) exp(-Cijdi/p) - 1) B i j e i e j t / ( A d i )(4)
+
where i = r/d
(5)
A = # exp(-d/p)
(6)
= #ij/$ - 1
(74
Aij
-1
(7b)
= p/pij - 1
(7c)
B, = & j / {
c,
The superscript O in the above equations represents the potential for the hypothetical reference system in which the ions interact through soft-core and Coulomb potentials with the characteristic parameters $, 4, and p. The ions in the reference system have a unique mass, A, defined later by eq 24. #, 4, and p are obtained with eqs 4a-c in part 1, respectively. d is the characteristic separation distance determined with eq 8 in part 1. The momentum of an ion, pi, is reduced to a nondimensional form: where mi is the mass of ion i. The Hamiltonian of the mixture is expressed in a reduced form: A = H / A = A0 & (9)
+ A0 = cpi2/2 + 40 $0
I caoij i>j
& = A& = CCA$, i>j
(10) (11) (12)
H o is the Hamiltonian for the reference system and & the perturbed one arising from the perturbed pair potential.
pi
= (A/rni)1/2 - 1
(19)
d Qis the perturbed ter-m arising from the difference in the pair potential and hLmthat from the mass difference. hLmQis the cross term that arises from the difference in the pair potential and the mass difference. Collective transport properties, Le., electric conductivity, viscosity, and thermal conductivity, can be expressed with the help of the fluctuation-dissipation theorem: where A(t) is the dynamical quantity and A is its time derivative. The brackets ( )H represent the canonical average with respect to the Hamiltonian, H. The transport properties conjugated to A1T)'s are shown in Table I, with the reduced form of A(& A ( f )is divided into the reference term Ao(t^) and the perturbed term A(t^). These terms are also shown in Table I. Equation 20 is rewritten in a reduced form:
-
x( n)
K = KVkT A
= i m ( a ( 0 ) a ( n ) df fi
(21)
We chose a characteristic mass for the corresponding states correlation of the transport properties of. pure
1518 Ind. Eng. Chem. Res., Vol. 29, No. 7, I990
molten alkali halides in two ways in our recent paper (Tada et al., 1988): (1) mR-l = (mA-l mc-l ) / 2 (22)
+
ms-l12 = (mA-1/2+ mc-l12)/2
(11)
(23)
where mAand mc are masses of an anion and a cation of the pure salt, respectively. For pure salts, the corresponding states correlation for the electric conductivity has almost the same precision using either mR or ms as the characteristic mass, while that for the viscosity with ms was better than with mR (Tada et al., 1988). Thus, in this work we choose the following mass as the characteristic mass, rft, of the mixture, which reduces to ms for the pure salt: 2
fi-l/2
= C(xi/2)(mc;1/2 i=l
+ mAi-lI2)
(24)
where me and mAirepresent the masses of the cation and the anion in salt species i, respectively. We uselthe Liouville operator, eq 13, for the time evplution _of A and divide the dynamical quantity A into A' and d.( ) A rspresents the average for a system with the Hamiltonian, H. This term isexpanded around ( )fie with respect to _the perturbed AH. Thus, the autocorrelation function (A(Q)A'@)fi is expressed in terms of the referand the perturbed terms with ence term, (A(0)Ao(f))fio, respect to :he mass difference, K ~ the , pair potential difference, A& and both of them (see the Appendix). The lowest order of the perturbed terms with respect to Adij vanish due to the definition of +, 5, and p (eqs 4a-c in part 1). If the higher order terms with respect to A&, play a minor role in the autocorrelation function, the function can be expressed as (see Appendix) c.
['i(O)A(f)Ifi =
[AO(O)A(f)]fi,
+ 2i2,(@2n
+
n=l
where kois the reduced transport property for the reference system, whose ions have the uniqu: mass f i and interact through the reference potential 4'i,. (30)
3. Corresponding States of Transport Properties for Alkali Halide Mixtures The pair potential parameters +ij and pi, are taken from the report of Tosi and Fumi (1964), and the parameters ti,and Ci, are from that of Harada et al. (1983). The values of these parameters are given in Table I of part 1. The data of electric conductivity and viscosity of alkali halide molten salt mixtures are from Janz et al. (1974,1975,1977, 1979). The perturbation formula for the transport property was given by eq 29 in the preceding section. The pair potential in the form of eq 2 has two parameter sets, e2[/(Ad) and d j p . The transport properties should be reduced with these parameter sets. 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 e2[/(Ad) has a minor effect on the transport properties, and the reduced form of the tpnsport property, K, can be replaced b-y the simple form K*-as shown in Table I, by discarding V , T, and e2t/(Ad) in K. Tada et al. (1988) showed that the transport properties of the pure salts were correlated in the corresponding states without the scaled Coulomb potential parameter. The perturbation formula, eq 29, is rewritten as R * = -K- dZIi2 A' (fiii)'"
The first perturbed term involving 6 represents the effect of ionic mass difference. With eq 24, 6 is defined by 6 = 6C = -6, (26) 2
6, =
Cx,ct,,,
r=l
v =
C, A
(27)
This perturbed term originally involves the odd terms, i.e., the terms proportional to P + l ,which vanish by selecting f i as defined in eq 24. The second perturbed term involving w,k in eq 25 represents the effect of the mass difference and the difference of softness of the core repulsion, the third involving wSpk does that of the mass difference and the differences of softness and strength of the core repulsion, and the fourth involving o does that of the mass difference and the difference of the effective Coulomb potential. wE,Upk, and U$,pk are defined as follows: 2 *( =
ExtX,B&cl + MA,)
(28a)
1,1=1
2 Opk
= xxxix,Cr:(k3 11=1
WCpk
+ PAJ)
= EE:XJ,A,,Ct,k(P~I + PA^) hJ-1
Substitution of eq 25 into eq 21 yields
(28b) (28~)
x".xQ
...
* p k , k,WJ.pkl...WJpk"
k l ,...,k,-0
+ Q *(nWt"I
(31)
Generally, k * and K *B are given as functions of the reduced state variables, T (=kTd/(e25))and V ( = V / ( N A d 3 ) , NA = Avogadro's number). When the characteristic separation distance, d , is selected such that the Helmholtz energy of the mixture is equivalent to that in the system of the hard-sphere repulsion and the Coulomb potentials, it is reported in part 1 that the molar voIume and the surface tension of the alkali halide mixtures are well dessribed by the law of corresponding states and that the V along the saturated liquid line is expressed by a universal functionpf T. Thus, the reduced forms of the properties K * and K *o in eq 31 would mainly depend on T along the saturated liquid line. 3.1. Electric Conductivity. The plots of the reduced electric conductivity, ii* = Kd2(AA)'I2/e2,for pure molten alkali halides at any reduced temperatures provide parabolic curves with respect to 62. Thus, ii* of the pure molten salts is expressed as (Tada et al., 1988) i* = 9 *$ + *e464 (32)
s
+
s*,,= 1.79 X 10-3/p- 6.60 X
A*e,
= -1.68 X 10-3/f'
+ 2.72 X
s*,,fj4
(33) (34)
The plots of ii* - s*,,62 for the alkali halide mixtures at any reduced temperatures provide parabolic
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1519 0 LiF-LICl* 0 LiF-LiBrr 9 LIF-LI I *
LICI-NaCl* LIci*-tici H LICl-Rb(li* LiF-NaFr 0 NaC1-NaBrr e LIP*-KF Q NaCI-Nal* NaFx-KF 0 NaCl*-KC1 e KF-NaClr 0 NaCl-KBr* KF-KCl+ El NaCI-KI* 0 KF-KBrr N NaC1-RbCl* e KF-KI* KCI-NaBrx 0 LlC1-LIBrr KCI-Nal* Lici-Lii+ E KCl-KBr*
+
+
I
KCI-KIS
a KCI*-RbBr
0 RbCI-KBr* gl RbCI-RbBr* B RbCI-Rbl* ALiBr-Lilr A LiBr-NaBrr A LIBr*-KBr A LIBr-RbBrx A NaBr-NaI* V NaBrr-KBr 0 NaBr-RbBr*
V KBr-KI*
V
KBr-RbBr* RbBr-Rbl*
0 Li I-Nal* 0 0 0 0 0 D I
LI I*-KI L1 I-Rbl* Nal*-KI Nal-Rbl* KI-Rbl* NaF-NaCI* I
I
ucai~ C S/cm
1
Figure 2. Comparison of the electric conductivity calculated from eq 35, K ~ with ~ the , observed ones, K~~ The key is the same as in Figure 1.
O I
I
1
l/P
60
70
Figure 1. Corresponding states correlation for electric conductivity of alkali halide mixtures. * means the component, the mole fractions of which are shown in the figure.
curves against w p (=apl, eq 28b). ii* of the molten salt mixtures is expressed as ii* = ii*o 9 * , p 9*,,64 Q*,,w, Q*ezWp2 (35)
+
+
+
+
Q*,,= 7.96 X - 0.747 Q*,, = 3.14 X 10-3/i' - 3.12
(36) (37)
The reduced electric conductivi_tyof the-reference system, ii*o, is evaluated from (ii* - S *,,a2 - S *e4 h4 - Q * e l ~ p - Q *, up2).The resultant ii*o values are plotted against ? in bigure 1. ii*o is correlated by eq 38 with a 12% root-mean-square (RMS) deviation: ii*o = 0.208 exp(-0.0339/i') (38)
At x = 0.4, 0.6, and 0.5, the ?*is of systems LiF-LiC1, LiF-LiBr, LiC1-RbCl, LiBr-KBr, and LiBr-RbBr, which all contain lithium, do not fall near the solid line. Except for these systems, the RMS deviation is 8.9%. Equation 35 reduces to eq 32 for the pure molten salts because wp = 0 at x1 or x 2 = 0 by the definition of up. The coefficients in eqs 33,34, and 36-38 and also those in eqs 42 and 43 stated below are evaluated by least-squares fitting of the data. Figure 2 shows the comparison of the electric conductivity calculated from eq 35, K ~ with ~ the , observed one, K o b s , where the RMS deviation of K~~~ from Kobs is 15%. Equation 31 has other perturbation terms that contain ut, wtn (n 1 21, and (n 1 1). It theoretically cannot be said that these other perturbation terms are neglected
compared with the terms with w (=wpl). The reduced electrjc condu_ctivityfrse from &-andupperturbation terms, p - S * e262-S* 6 4 - Q * ,pp- Q *e2wp,was plotted with wE, %n ( n 2 2), or wJ.pn( n 2 1) at any reduced temperatures, T; however, any meaningful correlations were not observed. Thus, we concluded that the terms with wc, wpn ( n 1 2), or w~~~ (n 1 1)contribute little to eq 31 and that eq 35 gives the corresponding states correlation of the electric conductivity. The numerical values of the exponent and coefficient in ii*o (eq 38) and the function form of S and S (eqs 33 and 34) are different from those reported elsewhere (Tada et al., 1988) for the pure molten salts. The differences arise from using much more data in this wqrk than ip the previous paper. The functional forms of S and S presented in this work fit the data better than those in the previous paper. The value of ii*o for the pure molten salts ( x = 0 or l),however, is almost the same as that in the previous pfiper a t any reduced temperature in the range 40 C 1/T < 70. 3.2. Viscosity. The reduced viscosity q* = qd2/(hrii)1/2 for the pure molten alkali halides is expressed as follows (Tada et al. 1988): = -* + j * 62 (39) V O u2 S*,, = 0.366/? - 13.1 (40)
e*
The experimental values of q* - 9 *u$2 for the mixtures take quadratic form against wp at any reduced temperatures. Thus, q* of the mixtures is expressed as = + 9. u2 62 + Q * + Q*uzwp2 (41)
e*
Q*,,
= 10.6/? - 4.59 X lo2
Q*,,= -4.86
X
lo2/?
+ 2.17 X lo4
(42) (43)
(=e*
The redu-ced viscosity of the reference system,Afj*o S *$ - Q * w - Q *u2wp2),is plotted against 1/T in Figure 3. The ij*:;alue can be correlated by eq 44 with 12% RMS deviation. Equations 40 and 44 have been obtained for pure alkali halides (Tada et al. 1988). = 0.0608 exp(-0.0683/?) (44) Figure 4 shows the comparison of the viscosity calculated from eq 41, qcdc, with the observed one, qob. The RMS deviation of qcdc from qobs is 12%. As in the case of the electric conductivity, any meanipgful correlation? were not observed in the plots of fi* with w , w ,, (n 1 2), or ( nI S *uz62- Q *ulwp- Q 1)at any reduced temperature, +. 'fhus, we conclu ed that eq 41 gives the corresponding states correlation of the viscosity.
1520 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
- 2 Na
3,
&> I
a 1
3,
&'
- 301 , , , , 05 , , ,
2
I
,j 10
,;,05
-31, , , 0
'NaF
N
R
0 ,
,j 10
XL~F
1
2
i -4
i
Figure 5. Mixing electric conductivity of alkali halides. Figure 3. Corresponding states correlation for viscosity of alkali halide mixtures. The key is the same as in Figure 1.
'1,$id'
0
Figure 6. Comparison of the mixing electric conductivity calculated ~ that , from eq 55, A K ~with , the observed ones, from eq 46, A K ~and A K , ~ .The key is the same as in Figure 1.
and tiiare those for pure salt i. Figure 5 shows some representative examples of the mole Figure 4. Comparison of the viscosity calculated from eq 41,q d c , fraction dependences of A K , ~ and , ~ A K ~ In~ most . of the with the observed ones, qob. The key is the same as in Figure 1. systems studied, A K , ~ ' Sare reproduced by A Kas ~shown ~ in Figure 5a,b. In two systems, LiF-LiC1 and LiF-LiBr, 3.3. Mixing Quantities. The mixing electric conducthe absolute values of A Kare~much ~ smaller than those tivity, AK, is defined by of A K , as ~ shown in Figure 5c, and A Kof LiF-LiI ~ ~ does AK K - XlKl - X2K2 (45) not represent the feature of A Kas ~shown ~ in Figure 5d. The reason would be that the anions have relatively large where K is the electric conductivity of the mixture and K~ in lithium halide mixtures, the effect of which and K~ are those for pure salts 1 and 2, respectively. A K ~ polarizability ~ cannot be incorporated into the effective Coulomb pois expressed as follows: tential completely. These three mixtures include LiF and/or LiI, the correction factor kij of which is larger than AKcalc E Kcalc - XlK1,calc - ~ 2 ~ 2 , c a l c (46) those of the other salts as shown in part 1. however, does not change appreciably with kij in all the mixtures studied. The values of A Kare~compared ~ with those of A K , a~t x = 0.5 in Figure 6a. A Kagrees ~ ~with A K , considerably ~ well except for mixtures LiF-LiC1 and LiF-LiBr. This indicates that the mixing electric conductivity is represented by eqs 46-48. In a similar way, the mixing viscosity, Aq, is defined by
AV E t - nit)i - ~ AqdC
d and [ are the parameters for the reference salt, and dii
2
~
2
(51)
is expressed as follows: A ~ c a ~Ec Vcalc
- ~ i t l , c a ~-c~ 2 ~ 2 , ~ a l c
(52)
Ind. Eng. Chem. Res., Vol. 29, NO. 7, 1990 1521
'
(b)NaCi-NaBr
loa32 [ K I
1
(c)KCl-NaCl
1100[Kl
u1
-mobI L
Figure 7 shows the mixing viscosity, Atlob, and Avcdc. Avobs is reproduced well by Avcdc in KC1-KBr, while Aqdc shows the opposite sign to Aqoh in NaCl-KCl. In NaClNaBr, Aqob shows some scatter around zero, and it is not certain that A)lobs is reproduced by AqdC in this mixture. In the mixtures other than the three shown in Figure 7, the viscosity data of the pure salts ( x = 0 and 1.0) were not reported by the workers who reported the mixtures' data. Then, Alldc was compared with Aaoh obtained with the viscosity data of the pure salts reported by the other workers (Janz et al., 1974, 1975). A?calc agreed with the resultant Aqobs with almost the same precision shown in Figure 7a. AqdC changes little with kij in all the mixtures, as well as A K , , ~ ~ The values of Avdc are compared with those of AqOh at x = 0.5 in Figure 8a. The former agrees with the latter satisfactorily except for mixtures NaC1-NaBr and NaClKCl. Now AK,, which is a part of A K ~and ~ A?,, , a part of Aqcalc, are defined respectively by
where (57)
The values of AK, and A?, are shown by broken lines in Figures 5 and 7, respectively. AK, represents the main feature of A K ~and ~ A?, , does that of Avcalc. It can be said that AK, and As, are the main terms of A K and~ AT^^^^^, ~ ~ respectively. The values of AK, are compared with those of AKobs at x = 0.5 in Figure 6b. Although the values of AK, are slightly positive in mixtures NaI-RbI and KI-RbI, the mixing electric conductivity can be expressed by AK,, eq 55, satisfactorily. The values of A?, are compared with those of A?obs a t x = 0.5 in Figure 8b. The former agrees satisfactorily with the latter except for mixtures NaC1NaBr and NaC1-KCl. These indicate that the mixing electric conductivity and the mixing viscosity are strongly affected by up,written as eq 57, which is the parameter of the product of the difference of the ionic mass and that of the softness of the core repulsive potential. Equation 55 is a little troublesome to estimate the AK value, in which the parameter A = $ exp(-d/p) contains $, p, and d , which are expresged by eqs14a, 4c, 8, and 9 in part 1, and the coefficients Q *el and Q *e are expressed by eqs 36 and 37, respectively. A simpfer equation is desirable to estimate the AK values. AK in Figure 6a,b contains the data a t several temperatures. AKobs for the same key has almost a constant value except for LiF-LiI. ~ very small Then, the temperature dependency of A K , , is for almost all the mixtures. This indicate? that the temperature dependency of_d, A, Q *g, and Q in eq 55 is very small. Thus, d, A, Q *e,, and can be replaced by some constants. Here, d and A are replaced by charac-
d*,,
-
1
,
Y
Y
o
z
o
~
O
-
1 1 ' ' ' 10-02;
o ' 'z ' ' ~0 5
'KSr
'
'
'Nab
'
' 05 '
'
' ' 10
'NaCl
Figure 7. Mixing viscosity of alkali halides.
-0.2 -0.4 -0.4
-0.2
0 ~
0.2
0.4 -O-b.4
n CmPa.sl ~ ~
i
-0.2 AT^ 0 [ mPas 0.2 30.4 ~
Figure 8. Comparison of the mixing viscosity calculated from eq 52, A?calc, and that from eq 56, A?,, with the observed ones, A'lob. The key is the same as in Figure 1. Table 11. Characteristic salt LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr CsF CSCl CsBr CSI
Molar Volume and TemDerature Tij, K Vij,cm3/mol 1246 987 962 889 1168 1048 994 933 1084 1049 969 909 1044 963 911 876 919 897 928
14.60 28.83 35.15 44.43 20.74 37.09 43.18 54.05 29.71 48.91 54.89 66.19 35.52 52.98 72.81 39.65 59.85 67.11 82.02
teristic the molar volume, Vo,and characteristic temperature, To,of the mixture, which are defined by (58) 2
To 5 Z E ~ j x j T j j i. j = l
(59)
respectively, and the constants for Q *el and Q are determined below with the data of AKobs at x = 0.5. In eqs 58 and 59, Vij and Ti. are the characteristic molar volume and temperature of the pure salts, which were determined in the corresponding states correlation of the pure molten salts (Harada et al., 1983) and were given in Table 11. at x = 0.5 provide Plots of a parabolic curve with respect to u pregardless of temperatures. Thus, AKobs a t x = 0.5 is correlated with AK,' expressed by AK,' =
-2 c
( Vo/NA)2/3( k T 0 k )I2
(-4.62 x 10-3 +
9.67
X
10-3wp- 11.30,2) (60)
1522 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 ( d l ) g 0=
perturbed term of the first order with respect to
Aij,Bij, and Cij in the perturbation expansion of the Ham-
iltonian, defined by eq A8
iL = Liouville operator
Figure 9. Comparison of the simplified mixing electric conductivity which is calculated from eq 60, AK;, with the observed ones, A K , ~ . The key is the same as in Figure 1.
Equation 60 does not show zero at x = 0 or 1 due to the constant -4.62 X which arises fzom the neglect of the temperature dependency of d, A, Q * e l , and Q *e Comparison of h o b s with A K ~ ' at x = 0.5 is &own in Figure 9a. Figure 9b shows the comparison of AKobswith AK"' at x = 0.12, 0.2, 0.25, 0.37, 0.4, 0.5, 0.6, 0.63, 0.75, 0.8, and 0.88. AKobs is correlated with eq 60 at these mole fractions satisfactorily well except for LiF-LiC1, LiF-LiBr, and LiF-KF. From these figures, it can be said that eq 60 is the simplified expression of the mixing electric conductivity. In Figure 9a,b, mixtures that involve RbI are omitted due to lack of the values of Vijand Ti, for RbI in the report of Harada et al. (1983). For thermal conductivity, the corresponding states correlation was not obtained due to lack of the experimental data. However, a satisfactory correlation will be obtained for the thermal conductivity, and the mixing thermal conductivity will be correlated with up,by using the present approach. 4. Conclusion A law of corresponding states was developed for the transport properties of uniunivalent molten salt mixtures. The electric conductivity and the viscosity of the mixtures were correlated in corresponding states with four characteristic potential parameters and a characteristic mass. The mixing quantities of the electric conductivity and the viscosity were estimated with the corresponding states correlations and were strongly affected by a parameter, up, about the ionic mass difference and the difference of the softness of the core repulsive potential. The mixing electric conductivity correlation was reduced to a simplified correlation with the characteristic molar volume, temperature, mass, and parameter up. Acknowledgment
Y.Tada gratefully acknowledges the financial support from a grant-in-aid for fundamental scientific research, Ministry of Education, Science and Culture, Japan (63750894). Nomenclature A = dynamical quantity A V ) = time derivative of dynamical quantity A ( t ) AA(f) = preJurbed term of time delivative of dynamical quantity A ( l ) BIJ
=
$lJ/$
=
clJ/c-
-
c,, = PIP,, - 1
d = characteristic separation distance of the mixture e, = charge of the ith ion f = function H = Hamiltonian
K = transport property k = Boltzmann constant kij = correction factor of a pair between cation i and anion j (see eq 21 in part 1) mA = anion mass mA, = mass of anion in salt i mc = cation mass mci = mass of cation in salt i mi = mass of the ith ion mR = characteristic mass of pure salt defined by eq 22 ms = characteristic mass of pure salt defined by eq 23 fi = characteristic mass of mixture defined by eq 24 f i i = characteristic mass of pure salt i NA = Avogadro's number p i = momentum of the ith ion R~~= x component of momentum of the ith ion QEn = 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 Qdl,,,kn = reduced coefficient of the perturbed term with respect to the mass difference and the difference in the soft-core potential in the perturbation expansion of the transport property &nCo = 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 autocorrelation function fjpk,,,,k,(t) = reduced coefficient of the perturbed term with respect to the mass difference and the difference in the soft-core potential in the perturbation expansion of the autocorrelation function = ionic distance S, = reduced coefficient of the perturbed term with respect to the mass difference in the perturbation expansion of the transport property Sue( f ) = momentum correlation function between like-charged ions in reduced form, defined by eq A15 S,(t^) = reduced coefficient of the perturbed term with respect to the mass difference in the perturbation expansion of the autocorrelation function Sunlike(t^) = momentum correlation function between unlikecharged ions in reduced form, defined by eq A16 T = temperature TLj= characteristic temperature of the pure salt To= characteristic temperature of the mixture T = kTd/([e2), reduced temperature of the mixture p, = kTdii/(&ie2),reduced temperature of the pure sait t = time V = molar volume V l j= characteristic molar volume of the pure salt V, = characteristic molar volume of the mixture x i = mole fraction of salt i 2, = valence of the ith ion
Greek Letters P = l/(kT) A = perturbed term or mixing quantity 6 = aC = -bA = parameter of the mass difference defined by eq 2 1
cij = characteristic parameter of the pure salt
h = IC. exp(-dlp) 7 = viscosity of the mixture
vr = viscosity of pure salt i Avo = mixing viscosity calculated from eq 56 6' = reduced time 8, = reduced time K = electric conductivity of the mixture h i = electric conductivity of the pure salt
A K =~ mixing electric conductivity calculated from eq 55 AK/ = mixing electric conductivity calculated from eq 60 X = thermal conductivity of the mixture ph = parameter of the mass differenceabout the anion of salt
i wCI = parameter of the mass difference about the cation of salt i pl = parameter of the mass difference about the ith ion defined by eq 19 E = characteristic potential parameter of mixture = parameter of the effective Coulomb potential of the pure salt p = characteristic potential parameter of the mixture pl, = parameter of the soft-core potential of the pure salt T = reduced time = total potential = pair potential between i and j ions J. = characteristic potential parameter of the mixture $, = parameter of the soft-core potential of the pure salt wI = parameter defined by eq 28a up= parameter defined by eq 57 Wpk = parameter defined by eq 28b w$pk = parameter defined by eq 28c
[,
Superscripts m = perturbed term arising from the mass difference m+ = perturbed term arising from the mass difference and the difference in the pair potential = perturbed term arising from the difference in the pair potential = reduced form O = reference * = simplified form
+ A
Subscripts calc = calculated value e = electric conductivity like = like-charged ions obs = observed value unlike = unlike-charged ions u = viscosity 0 = reference
Appendix Time evolution of the reduced dynamical variable is expressed with the Liouville operator as
The first term in the right hand side of eq A6 is the reference term. The perturbation terms in eq A6 are classified into three groups, I, the perturbation terms arising from the difference of the pair potential, 11, those from the mass difference, and 111, those from the cross term of the pair potential difference and the mass difference. One of the first-order terms of the group I is taken for example: (*)~r.
=
(EXA&,)a i>j
The canonical average of a function f is expressed by the perturbation expansion with respect to the Hamiltonian difference:
1524
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 2
where
Cxl(pC1 + PAl) = 0
(A171
1=1
Then eq A13 is equal to zero. One of the second-order perturbation terms in group I1 is expressed as (
d1)& is the perturbation term of the first order with
respect to Aij,Bij,and Cij and can be rewritten as follows:
(dexp(tieo)aA)&= Jlike(i){(xlpC1+ x2c1C2)2 x2pA2)21
+ 24unlike(i)(xIpC1 + x2&3)(x1pA1
+ (x1pA1 +
+ xzpA2)
(A181
Substitution of eq A17 into eq A18 yields
$,(;)a2
(A19)
= 2{S^like(t^) - S^m]ike(i)l
6420)
( AA exp(ZiLO)AA)p =
where S^p(t^)
6 = XlYCl + X2PC2
(A211
In a similar way, it is shown that the odd terms in group I1 have the factor Cf=lxl(pcl+ p ~ l )which , vanishes, as well as eq A17 and the even terms have the factor J2". The first-order perturbation term with respect to i d m + in group 111 is expressed as follows: (AoJiexp{(I - O)iio)idmQ exp(Oiio)Ao 0
=
where the subscript "like" means the canonical average of the pair potential between like ions and "unlike" means that of unlike ions. Equations AlOa-c in part 1 are used for A,, B,, and C, in eq A9. Each term in the right-hand side of eq A9 vanishes due to the definition of $, p, and f (eqs 4a-c in part l), respectively, which are determined such that the free energy of the mixture interested is equal to that of the reference system. The first-order terms with respect to Ai,, B,, and C, in the other perturbation terms that arise from the potential difference vanish in a similar way. We assume that the higher order perturbation terms including two or more of A,, and C, in eq A7 contribute little to the autocorrelation function of the dynamical variable. Thus, group I vanishes in the perturbation expulsion, eq A6. The dynamical variable conjugated to conductivity is expressed as follows:
where
exp(6iL0)Aodo)
(A24) 8'
Ind. Eng. Chem. Res. 1990,29, 1525-1535
The nth order terms of way by (A0
X
S i0e x p l ( f
fW1exp{(B,,
k m 4 are
expressed in a similar
- 8 , ) i i D ) i d M S b01 e x p ( ( 8 ,- e,)iio)iaiW x ...
1525
by eq A31 from the results discussed above. This is eq 25 in the text.
(A(O)A(t,,,
=
( A ~ ( O ~ A ~ ( P+)n$3,,(2) ,i ,l
h2"
+
- 8,)iLo}idm+ exp(8,iio)Ao do1 doz ... d0,)fi. DD
Literature Cited where
exp(8,iLo)Aode1 ... de,)
(A30)
P
Here we assumed that the cross terms between wpk, u$ k, and ut play a minor role in eq A28 and are able to Le neglected. If we assume that the perturbation terms other than eq A28 in group 111, which include the perturbation factors i d # , iUm, AA, and & other than i d m @play , a minor role in the autocorrelation function of the dynamical variable, eq A6, the autocorrelation function is expressed
Harada, M.; Tanigaki, M.; Tada, Y. Law of Corresponding States of Uni-univalent Molten Salts. Ind. Eng. Chem. Fundam. 1983,22, 116-121. Janz, G. J.; Gardner, G. L.; Krebs, U.; Tonkins, 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 (fluorides and mixtures). Janz, G. J.; Tomkins, R. P. T.; Allen, C. B.; Downey, J. R., Jr.; Gardner, G. L.; Krebs, U.; Slinger, 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 (chlorides and mixtures). 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, 409-596 (bromides and mixtures; iodides and mixtures). 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 (mixed halides). Rice, S.A. Kinetic Theory of Ideal Ionic Melts. Trans. Faraday SOC. 1962,58, 499-510. Tada, Y.;Hiraoka, S.; Uemura, T.; Harada, M. Corresponding States Correlation of Transport Properties of Uniunivalent Molten Salts. Ind. Eng. Chem. Res. 1988,27, 1042-1049. 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. 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. Chem. Fundam. 1971,10,418-423.
Received for review March 6 , 1989 Revised manuscript received January 2,1990 Accepted January 25, 1990
Quaternary, Ternary, Binary, and Pure Component Sorption on Zeolites. 1. Light Alkanes on Linde 5-115 Silicalite at Moderate to High Pressures H.B. Abdul-Rehman, M. A. H a s a n a i n , a n d K.F. Loughlin* King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
Pure component and multicomponent equilibrium data are reported for the adsorption of the first four n-alkanes on Linde S-115 silicalite in the temperature range 275-350 K and a t pressures up to 1.723 MPa. T h e intrinsic Henry constants and the heats of adsorption are extracted from the data by using virial isotherm techniques. The data were modeled by using five isotherms explicit in partial pressure, four simple isotherms (LRC, Toth, Mathews and Weber, Jaroniec), and one statistical thermodynamic isotherm (Ruthven). The Toth model was observed t o fit the data best for both the pure component and multicomponent data. The Toth model reduces to the Langmuir isotherm for methane and ethane except a t 275 OK for the latter. In the demethanization of natural gas, a process stream consisting of methane, ethane, propane, and n-butane
* To whom correspondence should be addressed. 0888-5885/90/2629-1525$02.50/0
arises in which methane is generally separated from the other constituents by cryogenic distillation. Economic comparisons of cryogenic distillation with adsorption for purification involving light gases indicate that the com0 1990 American Chemical Society
