The Solubilities of Hydrogen, Deuterium, and Helium in Molten Li2BeF4 Anthony P. Malinauskas* and Donald M. Richardson Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830
Solubilities of Hz, Dz, and He in molten Li2BeF4 have been measured. Values of the Ostwald coefficient Kc (equilibrium ratio of gas concentration in the dissolved state to the gas-phase concentration) which were determined for He can be described over the temperature range 773-1073°K by the expression 0.7954, whereas the combined data for H2 and D2 can be reprelog (103K,) = log T - (1177/T) sented over the temperature range 773-973°K by the expression log (103Kc) = log T (1535/T) 0.7684.
-
Studies of the solubilities of gases in molten salts are sparse, and unresolved discrepancies between different investigators, as already pointed out by Paniccia and Zambonin (19721, are not uncommon. Moreover, it is not possible to resolve these discrepancies by appealing to theory, since an adequate theoretical treatment has yet to be developed (Lee and Johnson, 1969; Veleckis, et al., 1971). In addition, it is not a t all clear how some of the theoretical approaches that have been developed can be modified and extended to include molten salt mixtures. Such mixtures have become increasingly important solvent and heat transfer media. The molten compound LizBeF4 (Romberger, et al., 1972) appears to offer some advantages over alternate materials as a heat transfer fluid for both nuclear (Briggs, 1971-1972) and thermonuclear (Grimes and Cantor, 1972) applications; and in connection with both of these uses, it is desirable to determine the solubilities of the hydrogen isotopes in this fluid. Although Watson, et al. (1962), had made gas solubility measurements with a closely related eutectic some years earlier, as part of perhaps the most systematic study of gas solubility in molten salts that has been performed to date, their studies were restricted to the noble gases He, Ne, Ar, and Xe. The present investigation essentially duplicates (and confirms) the previous He results and extends the work to include HZand Dz. Experimental Aspects Interest in the solubilities of the hydrogen isotopes in molten salt media is partly due to the observation that these species can readily permeate through metals a t elevated temperatures. However, this capability complicates experimental determinations of the solubilities of these gases, so that a specially designed apparatus is required. The apparatus and the experimental procedure that were employed in this work have been described previously (Malinauskas, et al., 1972). The apparatus consisted of two double-walled vessels which were positioned within separate furnaces and could be isolated from one another, or interconnected, through the manipulation of a freeze valve. In brief, the experimental procedure involved the saturation of the molten salt in one of the two vessels (the saturator), followed by transfer of a known volume of the saturated fluid to the second vessel (the stripper) where removal of the solute was effected by sparging the salt with xenon. After a partial separation had been achieved by freezing the xenon in a cold trap, the gas was transferred to a volumeter for measurement, and subsequently a sample was taken for mass spectrometric analysis. The helium, hydrogen, and xenon used in this study were specified by the suppliers to be at least 99.9% pure, whereas the deuterium supply was determined in this laboratory to contain 99.7 atom % D and the remainder vir242
Ind. Eng. Chem., Fundarn., Vol. 1 3 , No. 3, 1974
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tually all H. Except for the xenon, the gases were admitted into the apparatus after passage through liquid nitrogen-cooled coils of copper tubing. The xenon, on the other hand, was recycled; the occasional replenishments that were required were taken directly from the high-pressure cylinder supply. As indicated in previously reported work (Malinauskas, et al., 1972), it was necessary to operate the diaphragm pump, which circulated the sparge gas through the salt, under a dry argon atmosphere. This gas, also specified to be a t least 99.9% pure, was taken directly from a highpressure source. Analysis of a sample of the LizBeF4 used in the investigation yielded the results presented in Table
I. Experimental Results Previous studies by Malinauskas, et al. (19721, indicated a linear dependence of solubility on saturation pressure for the systems under consideration. Similarly, this Henry’s law behavior was observed by Watson, et al. (1962), for the solubilities of helium, neon, argon, and xenon in a closely related solvent (a 64 mol % LiF-36 mol 7’0 BeFz eutectic). All of these measurements, like the present work, involved saturation pressures in the range 1-2 atm. Thus, Henry’s law was assumed to be valid in the data reduction. The application of Henry’s law for the solubility of hydrogen, rather than a square-root dependence which is characteristic of hydrogen solubility behavior in metals, is of course an indication that the dissolved species is molecular hydrogen. As is indicated later, dissolution of the hydrogen in molecular rather than atomic form is likewise supported by the relative solubility of this gas compared with the monatomic gases. Table I1 is a summary of the solubility data obtained in this investigation. For this purpose the solubilities are expressed in terms of the Ostwald coefficient K,, which is defined as the ratio of the dissolved-gas concentration to the gas-phase concentration. This quantity is related to the so-called Henry’s law constant Kh through the expresTable I. Composition of the Li2BeF4
Weight %
a
Theoret
Obsd
Li Be F
14.0 9.1 76.9
Fe
...
13.8 9.5 76.7 21Q
Ni Cr
... ...
S
...
Parts per million.
86
61a 3a
Table 11. Solubilities of Helium, Hydrogen, and Deuterium in Li2BeFI
T,OK
He
H?
Dz5
"Hydrogen"
773 873 973 1073
3.86 i 0.25 (4)c 6 . 0 1 i 0.11 ( 8 ) d 9 . 1 6 i 0.18 (4) 1 4 . 5 1 i 0.22 ( 5 )
1 . 1 3 =k 0.08 (4) 3.17 i 0.09 (7)d 3.87 i 0.37 (8)
1 . 4 1 =t 0 . 0 8 (5) 2.74 =t0.16 (5) 4.26 i 0.44 (7)
1 . 2 8 f 0.15 (9) 2.99 =t 0 . 2 1 (12) 4.06 i 0 . 4 5 (15)
b
Includes HPand H D also collected. H 2and D? results considered identical. Numbers in parentheses indicate number of measurements. Includes data reported earlier by Malinauskas, et al. (1972). sion
Table 111. Thermodynamic Variables Associated with the Solution of Gases in LiF-BeF2 Mixtures at 1000°K
K, = K & T in which ideal gas law behavior is assumed. Each of the tabulated values is an average of the number of determinations identified in parentheses; the uncertainties listed correspond to the average absolute deviation from the average value. The deuterium solubility determinations were complicated by an unexpected exchange reaction involving an unidentified source of hydrogen. This phenomenon was manifest in the appearance of significant amounts of H2 and HD in the gas collected in the volumeter during the deuterium solubility determinations. As a result, we assumed no isotope effect on solubility in the discussions which follow; i , e . , we adopt D2 solubility results which were calculated by considering all of the hydrogenic species that were collected in the volumeter to be equivalent to D2. A comparison of the D2 results calculated on this basis with the H2 solubility values indicates that, within the limits of experimental uncertainty, this assumption leads to no inconsistencies. Linear least-squares analysis of the helium data yields
log (106Kh)= 2.2905 - (1177/T) (2) as an analytical representation of the temperature dependence of the helium results, whereas a similar analysis of the combined data for hydrogen and deuterium yields log (1O8Kh)= 2.3175 - (1535/T) (3) In terms of the Ostwald coefficient, the corresponding equations are log (103K,) = log T - (1177/T) - 0.7954 (2a) for the helium data and log (103K,) = log T - (1535/T) - 0.7684
(3a) for the hydrogen results. The helium results of Watson, et al. (1962), can be described by the relation log (1OBKh)= 2.3437 - (1131/T)
Gas
TASz,
AHi,
kcal/mol
Hen 5 . 1 8 i 0.20 He* 5.39 i 0 . 4 9 Ha,D?bVc 7.02 i 1 . 8 0
kcal/mol
AS", eu
8 . 5 7 i 0.05 9 . 0 3 i 0.13 10.54 + 0 . 3 4
-3.4 i0 . 3 - 3 . 6 i. 0 . 6 - 3 . 5 IJ, 2 . 1
Results obtained by Watson, et al. (1962), with LiFBeF2 (64-36 mole % ) eutectic. Data obtained in the present study with LizBeF4.c Assumes Hzand DPsolubilities t o be identical. of solution of a gas in a solvent becomes solely a function of solute-solvent interactions if the isothermal change in state is performed a t constant gas-component concentration; i. e . , for the process X(g,c,) = X(d,c,) (6) where x represents one mole of solute, cd is the concentration of solute in solvent, and g and d denote gaseous and dissolved states, respectively. If the entropy of solution for this process is denoted by A So,we can write
AS" = ( A H , / T ) - AS2 (7) in which ( A H l / T ) is the entropy change associated with the overall isothermal process
X g , cg) = X d , c d (8) where ce is the concentration of solute in the gas phase, and 4 S p represents the entropy contribution due to isothermal expansion of the gas X(g,c,) = X(g,cd (9) In terms of experimentally determined quantities, eq 7 takes the form a In Kh TAS" - - - -=- -RR T In K , a(l/T) since
+
(4)
this expression yields values which are 25-3070 larger than the results obtained from eq 2. Although the data suggest a systematic error between the two sets of values, the results are in essential agreement. Moreover, we do not believe the discrepancy to be due to the relatively slight difference in chemical composition of the two solvents employed, Le., 64-36 mol % LiF-BeF2 in the earlier study and 66-34 mol % LiF-BeF2 in the present investigation. The solubility values may be converted to a weight basis through the use of the solvent density values p which can be calculated from an expression proposed by Cantor (1969) p = 2.413 - (4.88 X 10-4)T (5) Discussion As Blander, e t al. (1959), have pointed out, the entropy
and TAS2 = -RT In K, The thermodynamic variables derived from the present data are summarized in Table 111; the uncertainties attending the results reflect the standard deviations associated with the corresponding In Kh us. 1 / T representation as determined using the linear least-squares procedure. The agreement between our results for helium and the results obtained by Watson, e t al. (1962), is well within the mutual limits of uncertainty. The present data do in fact support the observation made by the previous investigators (Watson, et al., 1962) regarding the unusually large A S o values associated with solution of gases in LiF-BeF2 mixtures. Our results for hydrogen likewise indicate a large entropy of solution, but this observation must be qualified in view of the large error limits. Ind. Eng. Chem., Fundam., Vol. 13, No. 3, 1974
243
rs, A
rc, A
Table V. Comparison of Size Parameters in Describing Gas Solubility in a LiF-BeF2 Eutectic in Terms of the Uhlig Model
1.29 1.39 1.48 1.71 2.03
1.79 1.6
Gas
Rb
R,
RC
He Ne Hz Xe
0.50 0.65 0.65 1.53
0.57 0.66 0.75 1.41
0.87 0.69
Table IV. Size Parameters of Gas Molecules Gas He Ne
H* Ar
Xe
rb,
A
1.11
1.27 1.27 1.57 1.94
...
1.92
2.18
Many attempts have been made to account for gas solubility behavior in terms of the properties and interactions of the solute and solvent molecules; all of these approaches are essentially modifications of the “hole” or “scaled particle” description which appears to have originated with Sisskind and Kasarnowsky (1933). Thus far, however, none of the treatments have proved to describe the experimental data accurately. Moreover, it is not a t all clear how even a suitably modified treatment could be extended to include solvents which are mixtures. Accordingly, we have not attempted an exhaustive examination of the various theoretical approaches using the present solubility values. One approach which can be readily extended to include mixture solvents, primarily because it ignores a detailed consideration of interaction effects, is the theoretical description of Uhlig (1937) as modified by Blander, et al. (1959). By considering the isothermal processes involving (1) expansion of the gas from concentration c g to concentration cd, (2) contraction of the gas molecules to points and introduction of the solvent, and (3) subsequent expansion of the “dissolved” point molecules, one obtains the relation -RT In K, = E ( T ) 4 a N r C 2 y in which E ( n is an unspecified interaction energy, N is the Avogadro number, y is the “microscopic” surface tension of the solvent (and assumed equal to its macroscopic value), and r, represents the radius of the cavity in the solvent that is necessary to accommodate a solute molecule. It is customary to employ crystallographic radii of the gas molecules for the cavity size r, and to completely ignore the interaction energy E(??. However, if this is done, and the radius values reported by Huheey (1968) are used, eq 13 predicts neon to be more soluble than helium in a given solvent. This is contrary to observation, a point which appears to have been overlooked by recent investigators (Field and Green, 1971; Paniccia and Zambonin, 1972) and which appears to us to be significant. Table IV lists r,, the values presented by Huheey (1968), as well as two alternate sets of size parameters. One of the latter, which is denoted as Q, is determined using the relationship
+
r b = (3b/16aN)’i3 where b is the “covolume” or “excluded volume” term in the van der Waals equation of state. This parameter is weakly dependent upon temperature; the values listed are average values which were determined for the temperature range 500-800°C using the data tabulated by Hirschfelder, et al. (1954). The second alternate set of size parameters involves the radius, r,,. This parameter corresponds to one-half the value u that appears in the Lennard-Jones (12-6) interaction potential
W )= 44(a/rY2 - ( ~ / r . ) ~ l (15) which is used to describe the potential energy of interaction $ ( r ) between two molecules as a function of separation distance r. Note that n corresponds to the separation distance for which the interaction energy is zero. Another 244
Ind. Eng. Chem., Fundam., Vol. 13,No. 3, 1974
Rexp”OOOC
...
1.29
0.67 0.78 0.79 1.22
Rexp’oO*C
0.67 0.80 0.79 1.18
size parameter that can be considered involves the separation distance r, at which the potential energy takes on its minimum value, - e . This parameter, however, is directly proportional to u , and for this reason will not be considered further. Specifically, for the Lennard-Jones interaction potential rm =
2l
%
additionally, except in the case of helium, r, does not differ significantly from r,. The tabulated values for r,, were also obtained from Hirschfelder, e t al. (1954). Although the values determined for A S o in this and the previous study of Watson, et al. (1962), suggest otherwise, we have made the customary assumption that E ( T ) is zero in comparing the suitability of the three different sets of size parameters. The resulting comparison is presented in Table V. For the purposes of this study, we have made the comparison using argon as a standard; thus the
R, [(r,)gas/(ri)~rlz (i = c,b,a) (17) are the squares of the ratios of the appropriate size parameters of gas and argon. By eq 13, under the assumption that E ( T ) is zero R, = In (KJgas/1n = Rexp (18) Two sets of Rexp values are tabulated, one for 500°C and the other for 800°C. Both sets of values were derived from the present studies and the results of Watson, et al. (1962). Also note that the hydrogen results have been adjusted by the same proportions that are required to bring our smoothed values for helium into perfect agreement with the values derived from eq 4. The inadequacy of the simplified Uhlig model, regardless of the size parameters employed, is evident. The model does appear to yield a proper ordering of solubility values using either rb or r,; however, the abnormally large effect of effective nuclear charge on the helium crystal radius does not apply in the case of solubility, a t least in the solvent under consideration. Acknowledgment All of the mass spectrometric analyses were performed by Loness Guinn, to whom we are most grateful. We also wish to acknowledge the many useful suggestions which were made by J. H. Shaffer. Nomenclature b = “covolume” or “excluded volume” term in van der Waals equation of state, cm3/mol cd = concentration of solute in solvent, mol/cm3 cg = concentration of solute in gas phase, mol/cm3 E(T) = energy of interaction between solute and solvent molecules, cal/mol AH1 = change in enthalpy for the dissolution process defined by eq 8, cal/mol Kc = Ostwald coefficient, dimensionless Kh = Henry’s law constant, mol/cm3 atm N = Avogadro’s number, molecules/mol R = ideal gas constant, cm3 atm/”K mol or cal/”K mol R, = square of the ratio of the appropriate size parame-
ters of gas and argon molecules as defined by eq 17, dimensionless Re,, = ratio of logarithms of Ostwald coefficients as defined by eq 18, dimensionless r = molecular separation distance, 8, rt, = radius of solute molecule as determined from van der Waals equation of state, 8, r, = radius of cavity in solvent, cm; crystallographic radius of solute molecules, A r, = molecular separation distance a t minimum potential energy of interaction, A ro = radius of solute molecule derived from LennardJones (12-6) potential energy function, A A S ' = entropy of solution for the process identified by eq 6, eu A s p = entropy of isothermal expansion of gas as defined by eq 9, eu T = absolute temperature, OK Greek Letters y = "microscopic" surface tension of solvent, dyn/cm t = absolute magnitude of the potential energy mini-
mum, ergs = solvent density, g/cm3 = separation distance a t which the molecular potential energy of interaction is zero, A @ ( r ) = potential energy of interaction at the separation distance r, ergs
Literature Cited Blander, M., Grimes. W. R., Smith, N. V., Watson, G. M..J. Phys. Chem., 63,1164 (1959). Briggs, R. B., Reactor Techno/., 14, 335 (1971-1972). Cantor, S.,U. S. Atomic Energy Commission, Report ORNL-4396, 174 (1969), Field, P. E.,Green, W. J., J. Phys. Chem., 75, 821 (1971). Grimes, W. R . , Cantor, S.,"The Chemistry of Fusion Technology," D. M . Gruen, Ed., p 161, Plenum Press, New York. N. Y., 1972. Hirschfelder, J. O., Curtiss, C. F.. Bird. R . B.. "Molecular Theory of Gases and Liquids," pp 250, 1110, Wiley, New York. N. Y., 1954. Huheey, J. E., J. Chem. Educ., 45, 791 (1968). Lee, A. K. K., Johnson, E. F.. Ind. Eng. Chem., Fundam., 8, 726 (1969). Malinauskas, A. P., Richardson, D. M.,Savolainen, J. E., Shaffer, J. H., Ind. Eng. Chem., Fundam., 11,584 (1972). Paniccia, F.. Zambonin, P. G., J. Chem. SOC.,Faraday Trans. 7, 11, 2083 (1972). Romberger, K. A., Braunstein. J., Thoma, R. E., J. Phys. Chem., 76, 1154 (1972). Sisskind, B., Kasarnowsky, I., 2.Anorg. Allg. Chem., 214, 385 (1933). Uhlig, H. H.,J. Phys. Chem., 41, 1215 (1937). Veleckis, E., Dhar, S. K., Cafasso, F. A., Feder, H. M.. J. Phys. Chem.. 75,2832 (1971). Watson, G. M., Evans. R. B., Grimes, W. R.. Smith, N. V., J. Chem. €ng. Data, 7, 285 (1962),
Receiuedfor reuiew November 1,1973 Accepted February 21,1974
p u
This work was supported by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation.
Thermal Conductivity of Binary Liquid Mixtures Mahesh P. Saksena* and Harrninder Physics Deparfmenf, University of Rajasfhan, Jaipur-302004, lndia
Assuming a quasi-lattice model for liquid mixtures, a theoretical expression for the cross coefficients is given, following the treatment of Horrocks and McLaughlin for a pure liquid. The appropriateness of the mixing rule and the proposed relation has been tested, with success, by calculating the thermal conductivity of five binary mixtures.
Introduction The subject of the thermal conductivity of liquids is of considerable interest due to its technological applications as well as the understanding it provides for the molecular processes. In recent years several review articles have appeared on the subject (Eyring and John, 1969; McLaughlin, 1964, 1969). Comparatively little work has been done on the thermal conductivity of liquid mixtures, which are equally important. Recently, Jamieson and Hastings (1969) have reported binary mixture data at 0°C for about 60 systems and this extensive work provides an opportunity to carry out further investigations in this field. These authors have also reviewed the existing methods of estimating the thermal conductivity of the mixtures and have shown that the NEL modified equation (NEL stands for National Engineering Laboratory, Glasgow, Scotland) and the modified Filippov equation (Filippov and Novoselova, 1955) are in best agreement with the experimental data. Both of these equations contain a constant whose value has to be empirically determined for the system under investigation. McLaughlin (1964) has also suggested a quadratic mixing law of the form
where x refers to the mole fraction, hl and X Z are the values of the thermal conductivity for the pure components, and A' is the effective cross coefficient. This equation appears to be most logical and has been found to be in satisfactory agreement with experimental data by the empirical adjustment of the cross coefficient (Jamieson and Hastings, 1969). We here intend to rewrite McLaughlin's equation in a more appropriate form and suggest a theoretical procedure for calculating the cross coefficient, using the analysis of Horrocks and McLaughlin (1960) for pure liquids. Development of Theory Horrocks and McLaughlin (1960), assuming a lattice structure for the liquid, derived a relation for the thermal conductivity of a liquid in which the excess energy due to the temperature gradient is transferred mainly by means of the vibrations of the molecules. There is also a convective contribution to the thermal conductivity, but it is found to be usually less than 1% of the total value. The resulting equations are Ind. Eng. Chem., Fundam., Vol. 13, No. 3 , 1974
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