THERMAL CONDUCTIVITY OF BINARY MIXTURES OF ALKALI NITRATES
725
Thermal Conductivity of Binary Mixtures of Alkali Nitrates
by John McDonald and H. Ted Davis' Department of Chemical Engineering, University of Minnesota, Minneapolis, Minnesota
(Received June 19, 1969)
Experimental data are reported for the thermal conductivity of binary mixtures of alkali nitrates in the molten state. The excess thermal conductivities are negative and increase as the square of the difference between cation radii. A simple solution model shows that the behavior of the excess viscosity and thermal conductivity can also be attributed to differences between ionic masses.
I. Introduction The purpose of this paper is to report measurements of the thermal conductivities of several binary alkali nitrate mixtures as a function of composition and temperature. The excess thermal conductivities are negative for all systems studied. Murgulescu and Zuca2 observed the same behavior for the excess viscosities of several binary alkali nitrate mixtures. Also reported in this paper are viscosities of potassium and sodium nitrate solutions. The excess thermal conductivities are plotted os. the difference between the radii of the cations in the mixtures. It is found that the excess thermal conductivities increase as the square of the difference between cation radii. On the basis of a simple solution model it is shown that the behavior of the excess viscosity and thermal conductivity can be attributed to differences between the masses instead of to differences between radii of the cations composing the mixture. Whether either the mass effect or size effect dominate cannot be determined in this work since the cation mass differences vary very similarly to the cation radius differences for the alkali nitrate systems. Systems in which the mass difference is important and the size difference is not, such as silver nitrate-alkali nitrate mixtures, thallium nitrate-alkali nitrate mixtures, etc., must be studied to sort out the effects. Such studies are underway now.
11. Experimental Technique and Results Several methods are available for the measurement of the thermal conductivity of alkali nitrates. Bloom, et made use of a concentric cylinder apparatus as did White and Davis3b recently. Turnbul14used a hot wire method, but White5 has claimed that this method gives erroneous results due to the electrical conductivity of the salt. This method does work successfully for electrically nonconducting fluids. A list of experiments performed using this method is given by Turnbull. Gustafsson, Halling, and KjellandeF have used an optical method for the pure alkali nitrates. In our experiments, we have made use of a concentric cylinder apparatus sketched in Figure 1. The inner
cylinder was made of silver and the outer cylinder of type 446 stainless steel. The gap holding the molten salt between the cylinders was 0.1 in. and the length of the inner cylinder 7 in. White5has shown that convective and radiative heat flows in such a cell are small. The total thermal end losses were kept below 5% by making use of a Mycalex cap at the base of the cell and a thin-walled stainless steel tube supporting the inner cylinder and housing the heater leads at the top. Two tubes from the cylinder gap, one from the base and the other from the top, enabled the cell to be filled and emptied, using a remote reservoir of molten salt and a simple vacuum-argon pressure arrangement. This arrangement enabled the cell to be left undisturbed throughout the total period of experimentation-a useful attribute in high-temperature work where corrosion of electrical leads and other delicate parts is occurring. Since the end losses were small, a first-order estimation of these losses was considered sufficiently accurate. Thus we can write
Q
=
Qi
+
Qz =
(C
+ Dx)AT
(1)
where Q is the total heat flow from the inner to the outer cylinder, Q1 is the heat flow through the solid end connections, and QZ is the heat flow through the salt. x is the thermal conductivity of the molten salt and AT the temperature difference between the two cylinders. The constants C and D were found by standardizing the cell with ethylene glycol and Dowtherm A. The temperature difference across the gap was found by means of chromel-alumel thermocouples linked to a Leeds and Northrup microvolt amplifier. Care was taken to shield the preamplification parts of the circuit electrically and thermally where necessary. The power (1) Aifred P.Sloan Fellow. (2) I. G . Murgulescu and S. Zuca, Electrochim. Acta, 1383 (1966). (3) (a) H. Bloom, A. Doroszkowski, and S. B. Tricklebank, Aust. J . Chem., 18,1171(1965): (b) L.R. White and H . T. Davis, J . Chem. Phys., 47,5433 (1967). (4) A.G.Turnbull, Aust. J,Appl. Sci., 12,30,324 (1961). (5) L. R.White, Doctoral Dissertation, Univ. of Minn., 1967. (6) S. E. Gustafsson, N. 0. Halling, and R. A. E. Kjellander, 2. Naturforsch., 23A,44,682 (1968).
Volume 74, Number 4
Februaru 19, 1970
JOHNRICDONALD AND H. TEDDAVIS
726 VACUUM
ARGON
PRESa SUR^t1 I I,
Table I: Thermal Conductivities of Pure and Binary Alkali Nitrate Systems" %
Salt
Li Na
c
100 100 25 50 75 25 50 75 25 50 75 25 50 75 50
Li Li Li Na Na Na Na Na Na
HEATER TUBE TO MELTING CUP
K K K
I
SUPRAMICA INSULATING CAP
Rb
Figure 1, Concentric cylinder apparatus.
xmix
- X ~ H I-
The Journal of Physical Chemistry
cs
X
13.30 11.36 8.04 6.28 6.57 4.07 6.77 9.17 9.01 IO. 50 11.61 4.83 6,46 8.32 7.71 8.16 7.73 6.20
75 50 25 75 50 25 75 50 25 75 50 25 50
Cs
cs
K K K
cs
cs
Cs Rb Rb Rb cs
+ 0.00502' + 0.00642' + 0.00872' + 0.00672' + 0.00192' + 0.00812' + 0.00412' + 0.00412' + 0.00752' + 0.00482' + 0.00342' + 0.00682' + 0.0064T + 0.00552' + 0.00452' + 0.00422' + 0.00722' + 0.00452'
a Compositions are in molar percentages. Conductivity units are 10-4 cal/cm sec deg, and temperature T , degrees. 2' < 460".
to the nichrome heater was calcuIated by measuring the voltage drop across the heater and a precision 1-ohm resistor in series with the heater. The cell was housed inside a furnace constructed from a design by Kleppa.' The simple calculation of the thermal conductivities was made possible largely by the low thermal end losses which made an exact knowledge of the cell geometry in these regions unnecessary and made a standardization procedure quite sufficient. We have obtained the thermal conductivity of binary mixtures of alkali nitrates for the cation pairs Na-K, Na-Cs, Li-Cs, K-Rb, and Rb-Cs. The experimental data have been fitted to a linear dependence of thermal conductivity on temperature by means of a least-meansquares calculation. A small nonlinear dependence if present cannot be distinguished due to the inevitable small devia,tions of data from the best fitting line, caused by consistent and random experimental errors. The thermal conductivities of the pure and binary systems are given in Table I. We have approximated the data of and of Gustafsson, et ~ l . by , ~a linear curve fitting. Their data and the corresponding linear fits together with those of White and Davis3band of this paper are given in Figure 2 for pure KNOa. The equations for the linear fits are given in Table I1 along with the standard deviations u of the least-mean-squares fits. As can be seen from Figure 2 and Table 11, our data are intermediate between those of the other authors and the standard deviation of our results is small. The thermal conductivities for the Li-Cs system at 450" are shown in Figure 3. For our purposes, the excess thermal conductivity of mixing (or deviation from "ideality") is defined by the equation AX =
Rb
cs
HERMOCOUPLE HOLDERS
SILVER
K
100 100 100
%
Salt
XZHZ
(2)
~~
~~
~
~~
Table I1 : Least-Mean-Squares Fit of Thermal Conductivity for KNOs Source
This work Gustafsson, et al. Bloom, et al. White
T , "C.
b
x(10-4 cal/cm
8.04 IO. 76 5.14 7.46
BOO
deg)
4-0.00877'" 4- 0.00222'
+ 0.01942' + 0.00882'
2 0.14 0.25 0.28 0.23
Standard deviation.
where xmixis the thermal conductivity of the mixture and xi is the mole fraction of the ith salt. As can be seen from Figure 3, which is typical for the systems studied, the values of Ax are negative. Palyvos, et aZ.,* have predicted a negative value of Ax for binary mixtures of argon, methane, krypton, and xenon in the liquid state. Experimental results for the thermal conductivity of liquid mixtures (usually organic) have been obtained by Filippov and Novoselova.e The excess thermal conductivities that they report are also negative. In Figure 4,the values of the excess thermal conductivities of mixing are plotted vs. the mole fraction of the salt having the heavier cation. Smooth curves have been drawn through the experimental points by sight with the tacit assumption that the points lie on almost parabolic curves. All of the curves in Figure 4 are skewed toward the lighter cation-rich portion of the (7) 0.J. Kleppa, J . Phys. Chem., 5 9 , 175 (1955). (8) J. Palyvos, K. D. Luks, I. L. McLaughlin, and H. T. Davis, J . Chem. Phys., 47,2082 (1967). (9) L. P. Filippov and N. C. Novoselova, Vestn. Mosk. Univ. No. 3, 37 (1955).
THERMAL CONDUCTIVITY OF BINARY MIXTURESOF ALKALINITRATES
727
(3)
THIS WORK A GUSTAFSSON ETAL-----
-------
BLOOM ETAL
+ WHITE
where xi is the mole fraction of the ith component of the mixture, D is a quantity (perhaps temperature dependent) having the same value for all the mixtures studied, and XI is the sum of the cation and anion radii of the ith component of the mixture. A correlation similar to eq 3 was also observed by Powers, Katz, and Kleppa" for the volumes of mixing of the alkali nitrates. The form of eq 3 has been predicted by Reiss, Katz, and Kleppa12 by developing a conformal solution theory based on a simple potential model in which the shortrange repulsive interactions between like ions are ignored and the potential energy is assumed t o be pairwise additive with the pair potentials of the form
Y
0
w
./'
v)
I 13.0 -
YJ
,/'
a
0
2 12.0 x k
0
11.0
-
400 TEMPERATURE ' C
300
500
Figure 2. Comparison of thermal conductivity data of KNO, with earlier work.
where the form of fcrBdepends only on whether a@ is a like ion pair or an unlike ion pair. An example of the model defined by eq 3 is the Coulomb potential between like ions and the Coulomb potential plus a hard core cutoff at distance X for unlike ions. The RKK theory has been to predict that all the excess thermodynamic quantities of mixing are of the form illustrated by eq 3. Due to the fact that transport properties depend on the masses of the ions as well as the potential energy of interaction between the ions, there is no reason to expect that the excess transport properties can be correlated in
\
I
25
I
50 % c5
\\
I
75
3
Figure 3. Thermal conductivity of the (Li-Cs) NO3 system a t 450".
graph. The corresponding curves plotted by Murgulescu and Zuca2 for excess viscosities are less skewed. It should be noted, however, that the curve for which the experimental error is least important due to relatively large Ax, namely the Li-Cs curve, is fairly symmetric. Thus, if the excess thermal conductivities could be measured as accurately as the excess viscosities, the qualitative differences between the two quantities as ti function of composition might be reduced.
111. Correlations of Excess Thermal Conductivities and Viscosities with Molecular Parameters Kleppa and Hersch'O have found experimentally that the enthalpies of mixing of binary alkali nitrates obey the formula
VQ H E A V I E R
CATION
Figure 4. Excess thermal conductivity of binary mixtures of alkali nitrates a t various compositions. T = 450".
(IO) 0. J. Kleppa and L. 9.Hersh, J. Chem. Phys., 34, 351 (1961). (11) B. F.Powers, J. L. Katz, and 0. J. Kleppa, J . Phys. Chem., 66, 103 (1962). (12) H. Reiss, J. Katz, and 0. J. Kleppa, J . Chem. Phys., 36, 144 (1962). (13) H.T.Davis and J. McDonald, {bid., 48,1644 (1968). Volume 74, Number
4 February 19,1970
728
JOHNMCDONALD AND H. TEDDAVIS -frc ,052
-ft)
0.4
I
2
4 5
3
6
7
8 L'
0 0 MURGULESCUB Z U C A
e
MCDONALD a D A V I S
0 3.0
9 IO
,026
,070
,104
,130
0.6
0.0
1.0
-cs
' ( A x 1 Plot
0.3
0.2
Q
0.I
0.2
0,4
0.6
0.0
A i
terms of the length parameters alone. Nevertheless] Murgulescu and Zucaz have found that the excess viscosities of mixing of several binary mixtures of alkali nitrates obey the formula A7 = - z ~ z z ~ ( X ,
- XI)
(5) where a is a (temperature dependent) parameter having the same value for all the mixtures studied. Equation 5 is illustrated in Figure 5 in which we have plotted the e7cess viscosities of several equimolar binary alkali nitrate mixtures os. the difference of the radii of the cation components of the mixtures. Except for the point for the NaN03-KN03 mixture the data plotted in Figure 5 are taken from Murgulescu and Zuca's work. The viscosity of the NaNOa-KN03 mixture was studied in this laboratory and is presented in Table 111. Our data compare favorably with that given by Janz.I4 Table 111: Viscosity of (Na-K)NOa Mixtures a t T = 425' %'oK
100
0
75
25 50
50 25
75
0
100
7, CP
1.715 1.660 1.641 1,740 1.876
7 (ref
14)
1-72
1.87
In Figure 6, the excess thermal conductivities of the mixtures (at equimolar composition) studied in this work are plotted us. the parameter XZ - X.I The correlation is not as good as that for viscosity. A plot of Ax The Journal of Physical Chemistry
0.4
AA
Figure 5. Excess viscosity us. Ah and f,,for equimolar iiiixtures at 425'. Units of Aq are centipoises.
% Na
0.2
0
0
OR
(AX12
Figure 6. Excess thermal conductivities us. AX, (AX)2, and ft,for equimolar mixtures at 450". Units of An are lo-* cal/cm sec deg.
vs. (XZ - X1)21 also given in Figure 6, yields a better correlation, indicating that Ax is proportional to the square of XZ - XI in contrast to A7 which is linear in Xz - XI. This difference in behavior of A7 and Ax is surprising in view of the strong similarities in the dependence of the viscosity and thermal conductivity on the potential energy. This similarity is most clearly evident in the autocorrelation function formulas15for the transport coefficients. In view of the fact that A x and A7 correlate differently with respect to the parameter Xz - X1 and an intuitive feeling that a mass effect should be observed in transport phenomena] we introduce here an alternative correlation scheme for the excess transport quantities. If the kinetic contribution to transport is ignored (an excellent approximation for the coefficients of viscosity and thermal conductivity of liquids as dense as the molten salts) the coefficient of thermal conductivity (and of viscosity) of the pure ith salt can be written in the form #)
=
x*A(i)
+ xAc(i) + xCC(*)
(6)
where X A A ( ' ) represents the contribution to x from anionanion interactions] X C C ( ' ) that from cation-cation interactions. A contribution if it is assumed that energy is transported via a pairwise additive potential of interaction] will depend explicitly on the product N,NB (N, and No being the number of molecules of types a and Dl respectively) and will depend implicitly (14) G. J. Janz, A. T. Ward, and R. D. Reeves, "Molten Salt Data," Rensselaer Polytechnic Institute, Troy, N. Y., 1964. (15) R. Zwanzig, Ann. Rev. Phys. Chem., 16,67 (1965).
THERMAL CONDUCTIVITY OF BINARY MIXTURES OF ALIULINITRATES on composition through the effect of composition on the structure of fluid around the interacting pair aP. Next consider a mixture of charge symmetric salts 1 and 2 having a common anion A. The thermal conductivity of the mixture may be conveniently written in the form x(1,2)
+
729
For viscosity, our model solution assumptions take the form (I)
qab(lJ)
=
qas(l)
= Yep(2)
= xAA(1’2)f x1HAC(1’2) 22XAC’(1’2)f 212xcc(1*2)
+
+ 2X1x2xcC‘(1,2) (7)
(I)
xas(ll2)
= xas(l) = xa8(2)for all pairs a@
(y) V Z
2221(C!C’(1’2)
where xi denotes the mole fraction of the i t h salt and C and C’ denote the cations of salts 1and 2, respectively. I n writing the mole fractions in eq 7, we have again noted the explicit dependence of a term xas on the product NaN,+ I n a pure charge symmetric salt of N molecules, there are N anions and N cations. I n a binary mixture having a total of N molecules of charge symmetric salts having a common anion, there are N anions, x1 N cations of type C, and xzN cations of type C’, Equations 6 and 7 are general at this point. Indeed, they represent definitions of the quantities xas. We shall now introduce our model solution assumptions.
7lcc =
q*c
Again the mass dependence of qcct, etc., has been assumed to be the same as that predicted by simple molecular m0dels.’~*’7 Equations 14-17 lead to the following result for the excess viscosity
where
(8) The implications of our model are readily tested. since it can be readily shown that fx
HC’C’
=
1 (23mc,)l/z
I O andf,,
L: 0
First,
(20)
eq 12 and 18 predict, respectively x*c
where mc and mc? denote the masses of cations C and C‘, respectively, and X * C is a reduced quantity common to all pure and mixed alkali nitrates. I is an “ideal solution” assumption implying that the environment around an interacting pair aP is not very different from the corresponding environment in the pure salt. Since we are considering a common anion mixture, assumption I is probably not bad as a first approximation. Assumption 11, however, is more serious. It implies for one thing that the cations differ only in their masses, that is, that the differences in their cationic radii and dispersion interactions can be ignored. Moreover, we have assumed in I1 that the mass dependence of HCC’, xcc, and HCW is the same as that predicted by the kinetic theory of collisional transport in fluids of molecules interacting via simple model potentials, such as the hard core or square-well model potential^.'^^'^ On the basis of assumptions I and 11, the excess thermal conductivity becomes
where
(14)
Ax
5 0 and Aq L: 0
(21)
This prediction is consistent with the excess thermal conductivities reported here and the excess viscosities reported by Murgulescu and Zuca. Secondly, at a fixed composition plots of Ax and Aq vs. f, and f,,, respectively, for the various mixtures studied should yield straight lines. Such plots are shown in Figures 5 and 6. It is evident from these plots that the excess coefficients correlate about as well with respect to the mass parameters f, and f,, as they do with respect to the length parameter Xz and XI. Finally, our model implies that Aq and A x should be directly proportional to the product z1x2. This behavior is observed for Aq for all the systems studied by Murgulescu and Zuca and for the potassium-sodium system reported here. However, the plots of A x vs. composition shown in Figure 4 are skewed to the left of the predicted parabola, although for the system Li-Cs for which the experimental error is smallest compared to Ax, the plot is almost of the form of the predicted parabola. All in all, the agreement between the predictions of our model and experiment and the fact that Ax and Aq cannot be correlated with the same power of Xz - X1 (16) S. Chapman and T. G . Cowling, The “Mathematical Theory of Non-Uniform Gases,” Cambridge, London, 1964. (17) I. L. McLaughlin and H. T. Davis, J . Chem. Phys., 45, 2020
(1966). Volume 74, Number 6 February 19,1970
730
A. J. EASTEAL AND I. A I . HODGE
suggests that the differences in cation masses is at least as important as, and maybe more important than, the differences in cation radii in determining the excess transport quantities.
Acknowledgments. We are grateful for financial support of this research furnished by the National Science Foundation, the U. S. Army Research Office, Durham, and the Alfred P. Sloan Foundation.
The Electrical Conductance of Molten Lead Chloride and Its Mixtures with Potassium Chloride by A. J. Easteal'" and I. M. Hodgel" Department of Chemistry, University of Auckland, Aucklund, New Zealand
(Received July 81, 1069)
+
The specific conductances of molten PbClz KC1 mixtures have been measured as functions of temperature and ' PbC12, respectively. In the experimental temcomposition,for the ranges ca. 500-800" and ca. 20-100 mol % perature range the specific conductancesof all melts follow (a) quadratic temperature dependencesand (b) modified Arrhenius temperature dependenceswhich can be expressed by equations due to Cohen and Turnbull and Adam and Gibbs. Bases for the interpretation of the compositiondependence of conductancefor these mixtures a paramare examined, and the composition dependence is put on a rational basis in terms of the temperature TO, eter associated with the temperature of eero free volume, or zero configurationalentropy.
Introduction The composition and (to a lesser extent) temperature dependences of the transport properties of binary salt mixtures have been the subject of many papers in the molten salt field. Deviations from additivity relationships of, inter alia, conductance isotherms have been used diagnostically to infer the constitutions of many mixtures. Such deviations have led, for example, to the hypothesis that complex anionic species exist in molten lead halide-alkali metal halide and cadmium halide-alkali metal halide mixtures.'b However, published interpretations of isotherms of conductance and other transport properties are open to doubt on several grounds. I n particular, the absence of a suitable reference temperature for comparison of conductances makes deductions based on such comparison hazardous. It is not physically meaningful to ascribe differences in conductance of mixtures wholly to differences in the constitution of the mixtures, since the conductances may also differ because the comparison is made at nonequivalent temperatures. It has been suggested that the freezing point be used and that conducas such a reference temperature,1b$2 tances be compared at temperatures Or given by
er =
a T~
(1)
where Tf is the liquidus temperature of a given mixture and a is a constant assigned arbitrary values (e.g., 1.1, The Journal of Physical Chemistry
1.2,. . , ) However, the liquidus temperature of a given liquid mixture is determined at least as much by the structure of the solid phase formed on freezing as by the constitution to the liquid, so that Tr has no particular significance in respect to the transport properties of liquid mixtures. The critical temperatures would, perhaps, serve as reference temperatures, but again these are not governed wholly by the properties to the liquid phases and hence are still not entirely satisfactory. The development of liquid free volume and other theories of transport processes in liquids now indirectly provides a sound basis for comparison of the transport properties of mixtures of different composition. 3-10 (1) (a) Department of Chemistry, Purdue University, West Lafayette, Ind. 47907; (b) H , Bloom and E. Heymann, Proc. Roy. SOC.,A188,392 (1947). (2) I. S. Yaffe and E. R. Van Artsdalen, J. Phys. Chem., 60, 1125 (1956). (3) C. A. Angell, ibid., 68, 218, 1917 (1964); 69, 2137 (1965); 70, 2793 (1966). (4) R. Araujo, J . Chem. Phys., 44,1299 (1966). (5) M. H. Cohen and D. Turnbull, ibid., 31, 1164 (1959). (6) M . H. Cohen and D. Turnbull, ibid., 34,120 (1961). (7) G . Adam and J. H. Gibbs, ibid., 43, 139 (1965). (8) H. Bloom and A. J. Easteal, Aust. J . Chem., 19, 1577, 1779 (1966). (9) M. F. Lantratov and 0. F. Moiseeva, Russ. J . Phys. Chem., 34, 367 (1960). (10) C. T . Moynihan, J . Chem. Educ., 44,531 (1967) ; J.Phys. Chem., 70, 3399 (1966).
