Abstract
New enthalpy measurements above 2000 K have resulted in a complete reassessment and revision of the high temperature thermodynamic properties of hafnium. Values have been assessed to 4900 K and include a selected enthalpy of sublimation at 298.15 K of 621 ± 5 kJ/mol and a derived boiling point of 4850 K at one atmosphere pressure. A comparison is given between selected properties of the Group 4 elements to consider periodic trends.
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Introduction
Previous reviews on hafnium were by Hultgren et al.,[1] Spencer,[2] Gurvich et al.[3] and JANAF[4] whilst Onufriev et al.[5] reviewed values for the alpha phase. A further review is justified by including new drop calorimetry measurements above 2000 K determined by Rösner-Kuhn et al.[6] so that definite values can be calculated for the thermodynamic properties of the beta and liquid phases where previously only estimated values were available. As with zirconium, one of the major problems in carrying out experiments on hafnium is the avoidance of oxidation which could explain the rejection of a marked number of transition and melting points determinations as given in Tables 9 and 11 and the significant departure of specific heat and enthalpy values from selected values as given in Tables 12 and 13. A further complication for hafnium is that nearly all measurements were carried out on samples containing zirconium. Specific heat and enthalpy values were corrected using the Kopp-Neumann Rule as described in “Correction of Specific Heat and Enthalpy Values for Zirconium Content” section but this does introduce an increased uncertainty since this very often required an extrapolation of the zirconium values beyond their limits of phase stability.
From the evaluations in Tables 8 and 10, 2016 ± 20 K is selected for the transformation from the hexagonal close-packed alpha phase to the body-centred cubic beta phase and 2502 ± 20 K for the melting point. Wherever possible values have been corrected to the ITS-90 temperature scale[7,8] and to the currently accepted atomic weight of 178.49 ± 0.02.[9]
Alpha Phase
Roberts[10] selects 0.128 K as the superconducting temperature but there does not appear to be any specific heat measurements in the superconducting region. Therefore only the normal state is considered with the following values obtained for the electronic coefficient (γ) and the limiting Debye temperature (ΘD) as given in Table 1.
The selected values were combined with the specific heat measurements of Westrum[16] (5.8-349 K) to give values covering the range up to 298.15 K as given in Table 17. Additional specific heat determinations in the low temperature region by Cristescu and Simon[17] (13-210 K), Wolcott[11,12] (1.2-20 K) and by Burk et al.[18] (10-200 K) are compared with the selected values in Table 12.
In the high temperature region of the alpha phase, enthalpy measurements of Hawkins et al.[19] (339-1348 K) and Kats et al.[20] (1220-2001 K in the alpha phase) agree closely and also show a natural continuity from the low temperature data. The measurements of Hawkins et al. had already been corrected for zirconium content and were therefore only corrected for temperature scale (from IPTS-48 to ITS-90) but although the measurements of Kats et al. were on samples containing 0.78 wt.% zirconium no corrections were applied because of the very large extrapolation which would have been required for the alpha zirconium values. The combined values were fitted to the following equation with an overall accuracy of ±173 J/mol (1.33%):
The high temperature thermodynamic properties of all of the condensed phases are given in Table 19. The deviations of other specific heat and enthalpy alpha phase measurements from the selected values are given in Tables 12 and 13 respectively except for the specific heat measurements of Rumyantsev et al.[21] (1000-1900 K) which were given only in the form of a small graph.
Beta Phase
New drop calorimetry enthalpy measurements of Rösner-Kuhn et al.[6] (2096-2455 K in the beta phase) were corrected for 3 wt.% zirconium content using the values for beta zirconium selected by Arblaster[22] and fitted to the following equation with an overall accuracy of ±945 J/mol (1.37%):
The derived enthalpy of transition is 5040 ± 960 J/mol which can be compared with original experimental values given in Table 2. The derived specific heat is 37.15 ± 1.45 J/mol K.
The highly discrepant value obtained by Kats et al. is the main reason why their beta phase measurements were not included in the evaluation. The deviations of other specific heat and enthalpy beta phase measurements from the selected values are given in Tables 12 and 13 respectively.
Liquid
New drop calorimetry enthalpy measurements of Rösner-Kuhn et al.[6] (2339-2988 K in the liquid phase) were corrected for 3 wt.% zirconium content using the values for liquid zirconium selected by Arblaster[22] and fitted to the following equation with an overall accuracy of ±1691 J/mol (1.59%):
The derived enthalpy of fusion is 19885 ± 1937 J/mol which can be compared with original experimental values given in Table 3. The derived entropy of fusion is 7.95 ± 0.77 J/mol K and the derived specific heat is 36.26 ± 1.76 J/mol K
In addition, from entropy trends, Chekhovskoi and Kats[30] gives an equation which is equivalent to an entropy of fusion of 10.23 J/mol K for hafnium, equivalent to an enthalpy of fusion of 25,600 J/mol whilst Rösner-Kuhn et al.[31] estimates an entropy of fusion of 8.30 J/mol K equivalent to an enthalpy of fusion of 20,770 J/mol. The deviations of other specific heat and enthalpy liquid phase measurements from the selected values are given in Tables 12 and 13 respectively.
Gas Phase
Values are based on one bar standard state pressure and are calculated from the 466 levels given in Table 4 using the method of Kolsky et al.[32] and the 2010 Fundamental Constants[33,34] Thermodynamic properties of the monatomic gas phase are given in Table 20 and vapour pressure data in Table 21.
Enthalpy of Sublimation
For values given in Table 5 in the form of the Clausius-Clapeyron equation a “pseudo” Third Law value was calculated by evaluating the enthalpy of sublimation at the temperature extremes and averaging. The values obtained give a reasonable estimate compared to the values which would have been obtained if all of the data had been available. Because of a lack of detail as to what temperature scales were used in the measurements then no attempt was made to correct the values to ITS-90 from would be contemporary temperature scales.
Because of the close agreement between derived Third Law enthalpies of sublimation the measurements of Kibler et al.[39], Ackermann and Rauh[26] and Koch et al.[42] were accepted and averaged, with the assigned accuracy taking into account differences between ΔH(II) and ΔH(III) values. Because the other measurements showed poor agreement between ΔH(II) and ΔH(III) values they were rejected.
Vapour Pressure
The vapour pressure curve for the alpha phase as given in Table 6 was evaluated from free energy functions for the solid and the gas at 25 K intervals from 1400 to 2000 K and the transition temperature. For the beta phase the vapour pressure curve was evaluated at 25 K intervals from 2025 to 2500 K and the transition temperature and melting point and for the liquid at 50 K intervals from 2550 to 4900 K and the melting point.
Selected Condensed Phase Values at 298.15 K
Selection of the Transition Temperature and the Melting Point
Transition temperatures and melting points were corrected for atomic percent zirconium content based on the hafnium-zirconium binary system.[43] Separate correction values obtained for the solidus and liquidus equations were averaged as were values obtained for the alpha-beta and beta-alpha transition equations. Because of the provisional nature of the phase diagram, samples with high zirconium contents requiring large temperature corrections were not included in evaluating the selected values. Also because of a general lack of knowledge as to what temperature scales were used then only the selected values were corrected to ITS-90. Transition temperatures included in the evaluation are given in Table 8 and those not included in Table 9. Melting points included in the evaluation are included in Table 10 and those not included in Table 11.
Correction of Specific Heat and Enthalpy Values for Zirconium Content
Values for specific heat are reduced to J/kg K and for enthalpy to J/kg. Using the Kopp-Neumann Rule;
where X is the mass fraction of zirconium and C°p and (H°T−H°298.15) for zirconium are evaluated at the same experimental temperature as that of the alloy
A Comparison Between Experimental Values and the Selected Values
A Comparison Between the Present Evaluation and Previous Reviews
The boiling point calculated from the selected values of Gurvich et al.[3] at 4973 K at one atmosphere pressure is 123 K higher than the presently selected value of 4850 K whilst the boiling point given by JANAF[4] as 4964 K at one bar pressure is 119 K higher than the selected value of 4845 K. Since enthalpies of sublimation and the calculated free energy functions of the gas phase are similar in all three reviews then the differences are due to differences between the estimated values for the condensed phases in the previous reviews compared to the experimental values used in the present review. In the case of Gurvich et al. they estimated an enthalpy of fusion 6 kJ/mol higher than the present value and a specific heat for the liquid 8 J/mol K higher whilst JANAF estimated an enthalpy of fusion 9 kJ/mol higher although the selected specific heat of the liquid is only 1.5 J/mol K higher. However the result is that at 4900 K the free energy function of the liquid phase from the evaluation of Gurvich et al. is 2.55 J/mol K higher than the present value and from the evaluation of JANAF 3.33 J/mol K higher. The effect on the net free energy function is to lower the vapour pressure curve in both cases and lead to higher boiling points. It is therefore suggested that the much lower boiling point obtained in the present review is due to the use of actual experimental values.
A Comparison Between Group 4 Element Properties
The present evaluation for hafnium is combined with reviews on titanium by Desai[89] and on zirconium by Arblaster[22] and are summarised in Table 14. Since the three elements have similar electronic structures with the same sequencing of electrons in the outermost shells: Ti 3 s2 (10) 4 s2 (2); Zr 4 s2 (10) 5 s2 (2); Hf 5 s2 (10) 6 s2 (2) then this would suggest that the properties of hafnium could have been predicted. However this does not appear to be the case. The crystallographic and melting transition properties of titanium and zirconium can generally be considered to be similar but the values for hafnium show distinct differences mainly due to the marked increase in the stability of the hexagonal close-packed alpha phase. However in the case of the coherency of the condensed phases the enthalpy of sublimation and boiling points of zirconium and hafnium are very close whilst the values for titanium are different.
A Summary of Representative Equations
Specific heat, enthalpy and entropy equations above 298.15 K are given in Table 15. Free energy equations above 298.15 K in Table 16 and transition values involved in the free energy equations in Table 17.
Thermodynamic Tables
Low temperature thermodynamic data for the condensed phases is given in Table 18 and high temperature thermodynamic data for the condensed phases in Table 19. Thermodynamic properties for the gas phase are given in Table 20 and vapour pressure data in Table 21.
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Arblaster, J.W. Thermodynamic Properties of Hafnium. J. Phase Equilib. Diffus. 35, 490–501 (2014). https://doi.org/10.1007/s11669-014-0319-5
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DOI: https://doi.org/10.1007/s11669-014-0319-5