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J. B. Condon
Alternative Model for Nonstoichiometry in Uranium Hydride J. B. Condon Union Carbide Corporation, Nuclear Division, Oak Ridge, Tennessee 37830 (Received January 2 1, 1974; Revised Manuscript Received September 27, 1974) Publication costs assisted by Union Carbide Corporation, Nuclear Division
A model is presented to account for the nonstoichiometry and observed hydrogen overpressures for uranium hydride. The model assumes that vacancies in the hydrogen lattice are disfavored when the hydride is contaminated by oxygen and favored when certain metallic contaminants are present. This model explains the literature data as well as, if not better than, the model involving vacancy-vacancy attractions. I t is also better than the pure vacancy model. The presented model accounts for the observed hysteresis in the hydrogen pressure over uranium hydride and explains the first-order kinetics in uranium observed for hydriding.
I. Introduction This article reconsiders the basic model used to explain the nonstoichiometry of uranium hydride and the observed hydrogen pressures a t elevated temperatures. This reconsideration is prompted by the need to resolve a conflict that exists in the literature and to present a more consistent view of the kinetics and thermodynamics of this material. This consistency must be particularly applicable to recent findings with respect to contaminant effects. The literature presents considerable contradictory data on the equilibrium hydrogen pressure of the U-UH3-H2 system. Plateau pressure measurements by several authors are inconsistent. The most striking inconsistency, however, is in the full-phase diagram as presented by Libowitz and Gibbl and by Chevallier, et a L 2 The vacancy-vacancy model by Libowitz3 does not predict a critical temperature at 830’ described by the latter authors. This critical temperature is also inconsistent with the two-phase equilibrium pressures observed by Mallett and Trzeciak4 and reported to temperatures as high as 1400’. To resolve this conflict, additional considerations must be built into the statistical model for UH3. A further consideration is the recent findings of the effects of oxygen on both the kinetics5 and thermodynamics6 of this system. Experience with the kinetics of the uranium-hydrogen reaction cautions one to consider contaminant interactions, particularly interactions with oxygen. With sufficient oxygen contamination, the uranium-hydrogen reaction may actually cease far below stoichiometry. Oxygen background pressures as low as IOp7 Torr (-3 X 10-5 Pa) have profound effects on the kinetics and presumably the thermodynamics of the system. In a report, Meusemann and Von Erichsed conclusively demonstrate an effect of oxygen contamination on the thermodynamics of UH3. Therefore, special attention is given to the effects of contamination in the present statistical model. This model is, therefore, quite similar to that used5 to explain kinetics of the system as applied to the uranium side of the phase diagram. It accounts for the experimental data by Libowitz and Gibbl without assuming hydride vacancy-vacancy att r a c t i o n ~It . ~can also account for the data by Chevallier, et al., with the assumption of slight metallic contamination. The presented model by-passes some of the more questionable aspects of previous theories. One aspect is the asThe Journal of Physical Chemistry, Vol. 79, No. 1, 1975
sumption that at least the geometry of the metal lattice is unaffected by addition of hydrogen. This assumption allows the use of the van der Waal’s “equal areas” criterion to generate two p h a s e ~ . ~I>t sis obvious, however, that the transformation from a-uranium to @-uranium hydride changes this lattice in a gross fashion; the change is not isomorphous. The present model, therefore, assumes a-uranium and @-uraniumhydride are completely different phases. This not only frees the derivation from the assumption of vacancy-vacancy attractions to obtain phase separation (with a calculated critical point of -1413’) but also allows stoichiometry for the hydrogen terminal solubility and the point of the maximum concentration of vacancies to be unsymmetrical with respect to the center of the phase diagram. Furthermore, in contrast to the model by Libowitz, the present treatment does not require that the energy of formation of a vacancy to vary with temperature. The model by Libowitz is prominently used to substantiate the vacancy-vacancy attraction theory of ionic crystals as formulated by A n d e r ~ o n .The ~ model proposed in this paper, however, assumes no vacancy-vacancy attraction. Experimentally, there are enough conditions that may be varied to make a distinction. It is important to resolve this dilemma. However, resolution of this matter must await the accumulation of data a t higher temperatures and pressures. The concept of a mixed phase, with trace oxide ions behaving similarly to the hydride ions in the P-UH3 phase, is central to the proposed model. In terms of the U-P-UH3UO, ternary phase diagram (Figure l),this means that the single @-UH3phase not only extends toward the uranium corner of the diagram but also toward the uranium oxide corner as well. This extension of the single phase for a finite distance from the corner point is always implied, even if it is much too small to be measured experimentally. To assume otherwise would rule out the concept of solubility.
11. Theory Contaminant Modified. A. Symbols. The following symbols will be used in this discussion.
EI E V EVV
energy to form interstitial energy to form vacancies energy of vacancy-vacancy attractions
Alternative Model for Nonstoichiometry in Uranium Hydride
43
Ex
energy released in replacing a hydride ion by an oxide ion in the lattice k Boitzmann's constant Ki, Kz, K' temperature dependent constants stoichiometric number (i.e., UH,) number of hydrogen lattice positions (i.e., three times Avogadro's number per mole UH3) number of interstitial positions number of interstitial hydrogens number of sites the oxide ion can occupy number of hydrogen vacancies number of oxide ions replacing hydrogen hydrogen pressure oxygen pressure during hydride formation a positive number equivalent to average number of near-by hydride lattice positions affected by the presence of a contaminant (e.g., oxygen atom) such that vacancies will be unfavored (if favored q is negative) gas constant (8.314 J mol-I K-1 used here) (1 cal= 4.184 J) full stoichiometric number according to Libowitz absolute temperature canonical partition function for hydrogen in the lattice canonical partition function for uranium in the lattice canonical partition function for oxygen in uranium hydride lattice proportionality of the number of interstitial sites to the number of hydride ion positions negative deviation from stoichiometry ratio of oxide ion-to-hydrogen lattice positions activity of hydrogen activity of uranium activity o f oxygen a variable defined by eq 14 grand partition function
B. Vacancy Model Modified b y Oxygen Contamination. The model presented to account for the effect of oxygen on the kinetics of the hydrogen-uranium reaction5 assumes an interference by oxygen within the crystal structure of the metal. Hydrogen, occupying interstitial positions, diffuses rapidly in the metal before nucleation and growth of the /3phase uranium hydride. Oxygen or other anions compete for these interstitial sites and modify the kinetics considerably. The present model, which describes the hydride phase, makes a similar assumption about the role of oxygen in the crystal structure. This assumption is that oxygen in trace quantities and at a low activity can substitute for hydrogen in the anionic lattice positions and subsequently can affect the behavior of other hydride sites in the immediate vicinity. In the case of the uranium-hydrogen reaction, oxygen interstitials are established very slowly. This means that during the initial hydriding of metallic uranium the activity of the oxide ions is not equilibrated with the surroundings. It is reasonable to assume the same to be true for the measurements of Libowitz and Gibb. The greatest contamination by oxygen probably occurs during the creation of the hydride in the broad plateau region of the phase diagram. Knowing when this concentration is established is important in determining the energy of replacement of a hydrogen atom by an oxygen atom, since the replacement energy
Figure 1. Schematic of the hydride corner of the uranium-p-uranium hydride-uranium oxide ternary diagram illustrating a region for the hydride phase. Isotherms drawn illustrate the uncontaminated ws. contaminated case. The true contaminated isotherm which occurs at the dashed line labeled true composition line is shown projected back to the U-P-UH3 axis. This projection is labeled apparent isotherm and is compared to the isotherm that would be obtained in the absence of contamination.
and the oxygen activity appear in the same mathematical term in the present statistical model. It is reasonable to assume that the oxide ion in the lattice will affect the near-by hydride ions such that no vacancies will occur in those positions. This assumption originates from the fact that if the crystal structure of the uranium hydride is fixed then the primary determiner of the local electron densities is the uranium atom positions. The proton positions would be a secondary factor given their lower potential compared to the uranium cores. Surely, vacancies would be a very minor influence considering their low concentration. Thus, the creation of a vacancy would disturb the electron density very little in the vicinity of the vacancy; however, the absence of the proton would create a high effective negative charge. This would be very unfavorable in the vicinity of the localized negative charge associated with an oxide ion. The number of affected near-by positions will be left here as an arbitrary constant q. I t will also be assumed that an oxide ion replaces a single hydride ion in the hydride lattice. The statistical fit is not very sensitive to this latter assumption and an assumption of double substitution yields nearly the same result. Using single substitution and the parameter q , the grand partition function may now be written
The maximum term in Z m2y be obtained with respect to both N,and N I since they are in equilibrium at the time of the measurement. The maximum term in E with respect to Nx,however, is determined for a previous time when the activity of the hydrogen, AH, was a t the plateau pressure. The Journal of Physicai Chemistry, Voi. 79, No. 1, 1975
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J. B. Condon
relate to the time of hydride formation. Assuming N s one obtains
a
In [ m a term Z ] a AT,
=o=-
In N, -t
+ In xHZH- ( E J k T ) (3) is only dependent on the crys-
In (ND- N,) If N I
