296
GEORGEG. LIBOWITZ
ment is good in nearly all cases. The fact that Dawson’s values were obtained a t 18’ hardly affects the comparison, since the partial molal volume at high dilution is not very sensitive to temperature. We found practically identical values a t 13.8 and 25” in both CSa and CCL. We found evidence of secondary reactions in a drift with time in the brown solutions of iodine in methyl- and dimethylnaphthalene and in both of the silicones. The first group of 10 solutions, arranged in order of increasing solubility, are the nearly pure violet solutions, the ones which fall on the line when R(b In x z / bIn T) is plotted against R In x2 shown in the papers of Hildebrand and Glew2and of Shinoda and Hildebrand.4 The points for solutions in which solvation occurs fall below that line, because of the smaller entropy of solution. . The solubility is given as -log x2 in the second column of figures. In the third is given distance each point falls below the line in cal. deg.-l mole-’. This is essentially the amount by which the entropy of solution is diminished by solvation. We give it in this direct experimental form instead of attempting to convert it to entropy of transfer from hypothetical liquid iodine to solution, SZ - si, by applying the factor ( b In az/b In x 2 ) ~which , becomes somewhat uncertain as the solubility increases, as explained in the preceding paper. The value of Vz for all these solutions is greater than 59.0 cc. the (extrapolated) molal volume of liquid iodine at 25’. The connection between contraction and complexing was clearly set forth in the paper by Jepeon and Rowlin~on.~We differ from their interpretations only in one point. They inferred that their value, Vz = 67.9 in cyclohexane, the only violet solution they investigated, could be regarded as the normal liquid volume and that any-
Vol. 62
thing less indicates complexing. They suggest that pure iodine is not a strictly normal liquid. However, we find Vz = 60.7 cc. in CSz, a pure violet solution. It is to be expected, moreover, that fi2 would increase with decreasing solubility parameter of the solvent. This is confirmed by the range 60.7 to 100 cc. in the list of violet solutions in Table I. Contraction is strong in solutions in ethers and in the methylnaphthalenes and, as Jepson and Rowlinson found, in ethyl bromide and iodide, where both solubility and absorption spectra indicate strong complexing. The partial molal volumes in benzene, toluene, p-xylene and mesitylene, approximately 62 cc. , are less than in comparable, non-complexing solvents, such as cyclohexane, but larger than in the ethers. This may be interpreted as evidence that the solvation in these aromatics belongs to the class de.4gnated by Orgel and Mulliken’O as “collision complexes,” instead of definite stoichiometric complexes. The large entropy deficiency of solutions in the alcohols is not reflected in small values of Vz, but this is not to be wondered at, because complexing doubtless occurs at the expense of hydrogen bonds. The reader may make other comparisons. We offer one final comment. The “violet” solvents in Table I cover a 5-fold range in molal volume. If these liquids were quasi-crystalline lattices, the size of the “holes” they would contain would vary so greatly as to make it inconceivable that the partial molal vclumes of dissolved iodine would be so nearly the same as we find them to be. We gratefully acknowledge the support of this work by the National Science Foundation. (10) L. E. Orgel and R. S. Mulliken, J . A m . Chem. Soc., 79, 4839 (1957).
EFFECT OF THERMAL GRADIENTS ON METALGAS SYSTEMS’ BY GEORGEG. LIBOWITZ Atomics International, A Division of North American Aviation, Inc., Canoga Park, California2 and Contribution No. 2.47 f r o m the Department of Chemistry, Tufts University, Medford, Mass. Received September 18, 1967
The distribution of a gas dissolved in a metal under a thermal gradient and constant pressure is considered. An expression for the concentration gradient is derived. The effect of compound formation on the distribution and on the shape of pressure-composition isotherms is discussed.
Distribution of Gas in a Single Phase Metal System.-The relation between pressure, temperature and gas content in a metal-gas system exhibiting formation of a non-stoichiometric compound has been derived from statistical mechanical consideration^,^-^ and can be expressed in the form where p is the pressure of the gas a t any temperature T, and gas concentration n. n is expressed as ( 1 ) This research was supported by the Atomic Energy Commission. (2) Present address of author. (3)J. 8. Anderson, Proe. Roy. Soc. (London), A186, 69 (1946). (4) A. L. G.Rees, Trans. Faraday Soc., 60, 335 (1954). (5) G. G. Libowitz, 3. Chem. Phys., 27,,514 (1957).
the atom ratio of gas atoms t o metal atoms in the metal; s is the stoichiometric gas to metal ratio in the metal-gas compound; po is the equilibrium pressure of the two-phase region, the two phases being metal saturated with dissolved gas and the non-stoichiometric metal-gas compound; T, is the critical temperature or the temperature a t which the two phases become completely miscible. The equation will hold for the diatomic gas of any element although it is most often used for metal-hydrogen, metal-oxygen and metal-sulfur systems. Consider a metal bar of uniform density under a temperature gradient ranging from TOto T,. If n 12TcI.
For an infinitesimal volume in the metal bar, containing dn, gas atoms and dnM metal atoms a t an average temperature T, we can write dn, = dnMsp'/l exp[(m/T) - ( A / T ) ] (4) If the mean free path of the molecules in the gas
phase is less than the diameter of the vessel in which the gas is contained, then the pressure of the gas along the metal bar is essentially constant. Assuming a linear gradient from To to T,, then since the bar is of uniform density. N M is the total number of metal atoms in the bar. Putting this into eq. 4 and integrating gives s o N g d n ,= N N~sp'/* - ( T , - To)exp(Al2) X
ST:
exp(a/T)dT
n. Fig. 1.-Pressure-composition isotherms of metal-gas systems.
(6)
where N g = total number of gas atoms dissolved in the metal bar. The integral in eq. 6 can be evaluated numerically between definite limits 'To and T,, if the heat of formation of the compound, and the critical temperature are known. Denoting the integral by I , the equation can be rewritten exp(A/2) p'/2 = N ( T m - TO) sz
where N = N g / N ~ = average concentration of gas atoms in the bar. Substituting this into eq. 3 yields N ( T m - TO) exp(m/T) n = I
(7)
Equation 7 is an expression for the distribution of dissolved gas along a metal of uniform density under a linear thermal gradient. For gradients other than linear, the expression for the gradient corresponding to eq. 5 is substituted into eq. 4 and a similar analysis can be carried out. Equation 7 holds only for the solution of gas in metal and is no longer valid when the concentration is high enough to form a compound or new phase since eq. 1 is only applicable tp one-phase regions. Distribution of Gas in a Two-phase System.When the average concentration of gas in the metal exceeds a certain value, the metal-gas compound commences t o precipitate a t the low temperature end of the bar. In the compound, n is no longer negligibIe with respect to s, so that an
Temperature. isobar of gas in a two-phase system (metal and metal gas compound) under a thermal gradient.
Fig. 2.-Distribution
analytical expression for n cannot be obtained from eq. 1. A qualitative consideration of the distribu-
GEORGE G. LIBOWITZ
298
Vol. 62
were studied, a preliminary series of runs yielded isotherms in which the pressure in the two-phase (U metal and UHS) region did not remain constant but decreased with decreasing hydrogen composition thus leading to a sloping plateau region. A review of the literature showed that this phenomenon had also been found in the Pd-H,7-9 Cu-S,lo Fe-S,ll C O - S , Ti-S,la ~~ Zr-H,14 Th-H,15 rare earth-hydro0 0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00 gen?16 and Na-H1' systems. The sloping plateau Hydrogen composition-atom ratio H / U . regions were ascribed to impurities in some cases,16 Fig. 3.-Effect of a thermal gradient on uranium-hydrogen and to the rift theory1*of hydrides in others.l5 pressure-composition isotherms. Further investigation of the uranium-hydrogen tion, however, shows that the bar is divided into system revealed that a thermal gradient of 25" two distinct sections, consisting of a metal phase existed along the uranium sample. When this and a compound phase. This can be seen by con- gradient was removed, the pressure in the twosidering Fig. 1, which is a graphical representation phase region remained constant as shown in Fig. 3. of eq. 1. The set of pressure-composition isotherms The sloping plateau region was, therefore, a result shown is characteristic of metal-gas systems exhib- of the thermal gradient along the sample and can be iting formation of a compound or a new phase. explained on the basis of the metal to compound The regions of sharply rising pressure at the low phase boundary described above. As hydrogen and high composition ends of the isotherms corre- was added t o the uranium metal, it dissolved until spond to single phase metal and single phase non- the over-all hydrogen content exceeded No. Furstoichiometric compound, respectively. The con- ther addition of hydrogen resulted in the formation stant pressure portions, or "plateau" regions are of uranium hydride a t the low temperature end of characteristic of the coexistence of the two phases the sample. The equilibrium plateau pressure is in conformity with the phase rule. Consider the then determined by the temperature a t the metalconstant gas pressure above the metal bar to be P'. hydride boundary where the two phases coexist. I n traversing from the high temperature to the low As additional hydrogen was added to the sample, temperature end of the bar, it is seen that the gas more metal was converted to the hydride, and the composition increases, according to eq. 7, with the boundary moved toward higher temperatures, thus bar remaining in the metal phase until the tempera- increasing the equilibrium hydrogen pressure and ture Tz,where the solubility limit in the metal thereby leading t o a '(plateau" which sloped upphase is reached. Below this temperature only ward. Conversely, on dehydriding runs, as hydrothe compound phase exists. Therefore, both gen was removed from the sample the metal-hyphases coexist only a t the sharp boundary line be- dride boundary moved toward the cooler end of the tween metal phase and compound phase. The dis- sample thus decreasing the equilibrium pressure. tribution of gas in the bar can then be represented It is quite probable, therefore, that the sloping by the isobar shown in Fig. 2. plateay region in many of the systems mentioned The minimum over-all content of gas in the metal above also may have been due to thermal gradients bar necessary to form the compound phase a t the along the sample during the investigations. low temperature end can be calculated from eq. 7, (7) C. Hoitsema, Z . physik. Chem., 17, 1 (1895). if the terminal solubility of gas in the metal is (8) L. J. Gillespie and L. S. Galstaun, J . A m . Chem. Xoc., 18, 2565 known as a function of temperature. This mini- (1936). mum concentration, No, will then be given by the (9) L. J. Gillespie and W. R. Downs, ibid., 61, 2496 (1939). (IO) W. Bilte and R. Juza, 2. anoru. Chem., 190, 173 (1930). expression m
101
(11) R. Juza and W. Biltz, ibid., 206, 273 (1932). (12) 0. Hfilsmann and W. Biltz, ibid., 224, 73 (1935). (13) Biltz. P. Ehrlioh and K. Meisel, abid., 284, 97 (1937).
W.
-
(14) M. N. A. Hall, 9. L. M. Martin and A. L. G. Rees, Trans. Fara-
where nTo is the terminal solubility of gas in the metal a t To. Effect on Pressure-Composition Isotherms.During an investigation of the uranium-hydrogen system6 in which pressure-composition isotherms (6) G. G. Libowite and T. R. P. Gibb, Jr., THIEJOURNAZ. 61, 793 (1957).
day SOC.,4, 1306 (1944).
(15) M. W. Mallett and I. E. Campbell, J . A m . Chem. Soc., 7S, 4850
(1951).
(16) J. J. Kate and E. Rabinowitch, "The Chemistry of Uranium," McGraw-Hill Book Co., Inc., New York, N . Y . , 1951, p. 186. (17) M. D. Banus, J. J. McSliarry and E. A. Sullivan, J . A m . Chenz. Soc., 77, 2007 (1955). (18) D. P. Smith, "Hydrogen in Metals," University of Chicago Press, Chicago, Illinois, 1948.
Ir
