Abstract
The thermodynamic properties of tantalum have been evaluated to 5800 K. Selected values include an enthalpy of sublimation of 781 ± 4 kJ/mol at 298.15 K and a boiling point at one atmosphere pressure of 5762 K.
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1 Introduction
This work is one of a series on thermodynamic properties of the elements to appear in the Journal of Phase Equilibria and Diffusion. The articles are expected to be especially useful to those who are updating databases for phase diagram prediction and for other applications as well. Each article in the series will list the properties of a single element, including Ag,[1] Au,[2] Be,[3] Cr, Cu,[4] Hf,[5] Mo, Nb,[6] Ta, Ti, V[7] and W. References are given for papers which have already been published. In addition, the series will also include two summary papers on selected values of the melting points and boiling points of the elements and on the enthalpies and entropies of fusion and transition of the elements.
Inaba[8] determined the superconducting temperature to be 4.4924 K whilst a melting point of 3293 ± 15 K is selected as an average after correction to ITS-90 of values of 3292 K determined by Rudy and Harmon[9] and 3294 K determined by Pemsler[10] where in both cases the material used was of particularly high purity. Although Hiernaut et al.[11] initially also reported a similar value at 3290 K, later Hiernaut et al.[12] reported a much lower value of 3260 K without giving any explanation as to the large difference. Cezairliyan et al.[13] also selected a much lower melting point of 3258 K as a correction to ITS-90 of the value obtained by Malter and Langmuir[14] although no impurity content was given so that this value must be treated with caution. Rudy and Harmon[9] noted that increasing the total impurity content to only 500 ppm (0.05%) resulted in a 30 K drop in the melting point.
Wherever possible values have been corrected to the currently accepted atomic weight of 180.94788 ± 0.00002[15] and to the ITS-90 temperature scale using correction factors of Douglas,[16] Rusby,[17] Rusby et al.[18] and Weir and Goldberg.[19]
Based on dilatometric measurements, Bollinger et al.[20] suggested that all of the three Group 5 elements, vanadium, niobium and tantalum, underwent a martensitic structural distortion just below room temperature leading to a structure which was not cubic. However, although given in smooth form, neither specific heat measurements of Sterrett and Wallace[21] (12-543 K) or electrical resistivity measurements by Williams et al.[22] (80-400 K) showed any evidence of an anomaly that would be involved in a phase transition whilst lattice parameter measurements of Smirnov and Finkel’[23] established that the structure remained body-centered cubic down to at least 127 K. Therefore, until the observations of Bollinger et al.[20] are independently verified and the supposed new low temperature structure is fully characterized then it is assumed that the body-centered cubic structure exists over the whole temperature range (Fig. 1, 2, and 3).
Low temperature specific heat of solid tantalum, taken from Table 15
Specific heat of tantalum for 300 < T < 5800 K, taken from Table 16
High temperature thermodynamic properties of tantalum for 300 < T < 4300 K, taken from Table 16
Previous comprehensive reviews on tantalum were by Hultgren et al.,[24] Gurvich et al.[25] and JANAF (Chase[26]) and on the solid only by Maglić[27] and Bodryakov.[28]
2 Solid Phase
2.1 Superconducting State—Range 0 to 4.4924 K
In the description of the electronic structure of transition metals in terms of two electronic conduction bands then if the scattering of electrons between the two bands is sufficiently weak then both may separately undergo a transition into the superconducting state with the resultant formation of two energy gaps which is a Type II superconductor. Anomalies which appeared to confirm this behaviour in niobium were summarised by Sellers et al.[29] whilst, in particular, a similar anomaly was observed for tantalum in the specific heat measurements of Satoh et al.[30] However, based on specific heat measurements the anomalies for niobium were shown by Sellers et al.[31] to be due to hydrogen contamination and that with hydrogen removed niobium behaved as a typical Type I superconductor. On this basis, specific heat measurements of Shen[32] were selected for tantalum and below Tc/2, were Tc is the superconducting transition temperature, the electronic contribution to the specific heat (Ces) was fitted to the Bardeen–Cooper–Schrieffer equation which together with two lattice contributions leads to the full representation of the specific heat (Cs) below 2.25 K (Tc/2) as given by Eq 1:
where γ is the normal state electronic coefficient (6.02 mJ/mol K2) and Tc is selected as a working value of 4.467 K as an average of values determined for the samples I and II of Shen.
At Tc, a value of (Cs − Cn)/γ Tc = 1.53 is selected as an average of samples I and II of Shen[32] and when extrapolated to the selected value of Tc leads to Cs − Cn = 41.37 mJ/(mol K) so that Cs = 79.01 mJ/(mol K) at Tc. This value was combined with the value of 13.59 mJ/(mol K) at 2.25 K obtained from Eq 1 and an intermediate value of at the mid-point temperature of 3.369 K obtained by iteration so that at Tc the entropy values of the superconducting state and the normal state are equal. This gives Eq 2 for C s over the range 2.25 K to Tc:
Derived thermodynamic values based on Eq 1 and 2 are given in Table 14 whilst differences from the selected values of other measurements in this region are given in Table 10.
2.2 Normal State—0 to 298.15 K
Low temperature specific heat for tantalum in the normal state is given in terms of the modified Debye equation: Cp = D/T2 + γ T + A T3 + B T5 where D is the nuclear quadrupole coefficient, γ is the electronic coefficient and A and B are lattice contributions where A is usually represented in terms of a limited Debye temperature, θ D , where θ 3 D = (12/5) π4 R/A = 1943.770/A where R is the Gas Constant and A is given in units of J/(mol K4). For D, Shen[32] gives the value 0.01 mJ K/mol which is accepted. Leupold et al.[33] showed that discrepancies in the reported values of γ and θ D could be explained by an abrupt change in the slope of the specific heat values at 7.19 K which was due to the presence of a Kohn anomaly in the phonon density of states. Experimental values of γ and θ D are summarised in Table 1 where the subscripts γ1 and θD1 and γ2 and θD2 distinguish values above and below the transition temperature.
The measurements of Shen[32] and Leupold et al.[33] initially appear to disagree significantly with a preference being given to the values of Shen since the value of θ D at 258 K gave the best agreement with a value of 263.8 K selected by Alers[34] from elastic constant measurements. However, the two sets of measurements were reconciled as follows to cover the range up to 12.5 K:
Above 12.5 K and up to 298.15 K the specific heat measurements of Sterrett and Wallace[21] (12 to 543 K) were selected since these give the best agreement with the high temperature selected values. Because the values were given in a smooth form they were not separately fitted to analytical equations. Laser flash calorimetric measurements of Takahashi and Nakamura[35] (82 to 997 K) agree satisfactorily in the low temperature region trending from 0.1% high at 90 K to 0.7% high at 150 K to 0.4% low at 298.15 K. Thermodynamic values in the normal state below the superconductive transition temperature are also included in Table 14 whilst thermodynamic values at 5 K and above are given in Table 15. The deviations of other specific heat values in the low temperature region are given in Table 11.
2.3 Solid State: 298.15 to 3293 K
The requirement is not only to have a continuity with the low temperature specific heat curve but also to agree with the closely agreeing high temperature enthalpy values of Berezin and Chekhovskoi[36] (2440 to 3238 K) and Arpaci and Frohberg[37] (2653 to 3276 K in the solid range) in order to derive an enthalpy of fusion value which agrees with experimental values.
Specific heat measurements of Rasor and McClelland[38,39] (700 to 3200 K), Lehman[40] (100 to 3200 K) and Taylor and Finch[41,42] (100 to 3195 K) all showed a sudden sharp increase above 2800 K which was not shown in the measurements of Cezairliyan et al.[43] (1900 to 3200 K). Whilst attempts have been made to explain the sharp increase as real behaviour it is considered that it is simply due to sample oxidation or experimental error as evidenced by a similar sudden sharp increase observed in the complimentary thermal expansion measurements of Rasor and McClelland[38,39] when compared to the quality measurements of Petukov et al.[44] (1244 to 2267 K) and Miiller and Cezairliyan[45,46] (1500 to 3200 K). On these grounds, and because the first three sets of measurements differ significantly from those of Cezairliyan et al.,[43] then it is considered that the measured high temperature specific heat values of tantalum are unsatisfactory for representing the thermodynamic properties and therefore instead values are obtained by combining the agreeing drop calorimetry enthalpy measurements of Oetting and Navratil[47] (534 to 1383 K) (8 data points), Berezin and Chekhovskoi[36] (2440 to 3238 K) (11 data points out of 18) and Arpaci and Frohberg[37] (2653 to 3276 K in the solid state) (15 data points out of 19). The first two sets of measurements were directly corrected to ITS-90 whilst the measurements of Arpaci and Frohberg[37] were corrected to the selected melting point. Enthalpy measurements of Conway and Hein[48,49] (1315 to 2622 K) (12 data points out of 19) were also considered but were then rejected since they showed an average bias of 0.5% low compared to the selected equation. The selected values were fitted to Eq 6 which has an overall accuracy as a standard deviation of ± 310 J/mol (0.40%). The selection of this equation to cover the range from 298.15 K to the melting point gave the best continuity with the selected low temperature specific heat values:
Derived specific heat and entropy equations are given in Table 7. Thermodynamic properties derived from Eq 6 are given in Table 16 whilst deviations for specific heat and other enthalpy values are given in Tables 12 and 13 respectively.
2.4 A Comparison of Selected Values at 298.15 K
The comparison is given in Table 2. In the low temperature region above the superconducting temperature, Hultgren et al.[24] mainly combined the specific heat values of Sterrett and Wallace[21] and Kelley[50] whilst JANAF (Chase[26]) combined these two sets of measurements with those of Clusius and Losa.[51] Both Gurvich et al.[25] and in the present review preference is given to the measurements of Sterrett and Wallace.[21]
3 Liquid Phase
Enthalpy measurements of Arpaci and Frohberg[37] (3292 to 3358 K in the liquid state) were corrected to the selected melting point and fitted to Eq 7 with an overall accuracy as a standard deviation of ± 256 J/mol (0.18%):
The derived enthalpy of fusion at 33.883 ± 0.402 kJ/mol can be compared with the wide range of reported values given in Table 3, all of which, apart from the measurement of Arpaci and Frohberg,[37] were determined by rapid pulse heating. However, it is noted that the directly determined value of McClure and Cezairliyan[52] and the derived value of Pottlacher and Seifter[53,54] satisfactorily agree with the selected value.
Because of the very narrow 65 K range over which the drop calorimetry measurements of Arpaci and Frohberg were measured then the derived liquid specific heat is relatively poorly determined as 42.18 ± 4.62 J/(mol K). Other determinations of liquid specific heat used rapid pulse heating and are also given in Table 3 except for the measurements of Lebedev and Mozharov[55] which trend from 69.6 J/(mol K) at 3300 K to 27.7 J/(mol K) at 3900 K. It is considered that the measurements of Pottlacher and Seifter[53,54] at 40.9 J/(mol K) represents an improvement in the rapid pulse heating technique and show a relatively satisfactory agreement with the selected value. Thermodynamic values derived from Eq 7 are also given in Table 16.
4 Gas Phase
4.1 Thermodynamic Properties of the Gas Phase
Selected values are based on the 488 energy levels from the sources given in Table 4 with thermodynamic properties being calculated using the method of Kolsky et al.[56] and the 2014 Fundamental Constants (Mohr et al.[57,58]). Derived thermodynamic values are given in Table 17. Not included in Table 4 are eighteen levels given by Moore[59] which were shown to be spurious by Mocnik et al.[60] and a further five levels which were shown to be spurious by Jaritz et al.[61] It is noted that thermodynamic properties of the gas phase selected by both Gurvich et al.[25] and JANAF (Chase[26]) show differences from the presently accepted values which may be due to the fact that they selected only the energy levels of Moore[57] whereas the present evaluation included a further 228 levels.
4.2 Enthalpy of Sublimation
For values given 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 then averaging. Because of a general lack of detail as to what temperature scales were used and problems associated with the exact measurement of temperature then no attempt was made to correct vapor pressure measurements to ITS-90 from what would have been contemporary scales. However, the temperature values of Langmuir and Malter[62] and Fiske[63] were corrected from laboratory scales to IPTS-1948 using the method of Szwarc et al.[64] Values are summarised in Table 5. The selected value of 781 ± 4 kJ/mol is based on the close agreement between the values of Edwards et al.,[65] Gorbatyi and Shuppe[66] and the effusion measurements of Golubtsov and Nesmeyanov[67] with the selected accuracy taking into account the evaporation measurements of Golubtsov and Nesmeyanov[67] and the measurements of Langmuir and Malter,[62] although the latter showed a marked trend with temperature.
5 Vapor Pressure
The vapor pressure equations for the solid as given in Table 6 was evaluated from free energy functions for the solid and the gas at 50 K intervals from 1800 K to 3250 K and the melting point and for the liquid at 50 K intervals from 3300 to 5800 K and the melting point and were fitted to the following equation:
ln (p, bar) = A + B ln(T) + C/T + D T + E T 2
6 Comments on Previous Reviews of the Thermodynamic Properties
Gurvich et al.[25] also appeared to give a preference towards the high temperature solid enthalpy measurements, and the selected values agree fairly satisfactory with the presently accepted values showing a maximum specific heat difference of 2.7% higher at 2800 K. However, JANAF (Chase[26]) gave preference to the high temperature specific heat values of Cezairliyan et al.[43] and as a result selected enthalpy values are on average 1630 J/mol higher than the smoothed experimental values above 3000 K. In the high temperature region, Maglić[27] selected mainly the specific heat measurements of Lehman[40] and Taylor and Finch[41,42] which were rejected here for reasons given in Part 2.3. As a result, selected specific heat values trend from 0.9% low initially to 3.3% low at 1000 K to 16.8% high at 3200 K. Integration of the specific heat equation given by Maglić[27] led to marked enthalpy differences compared to selected values trending 1950 J/mol to 3070 J/mol higher in the narrow range 3000 K to 3200 K. The review by Bodryakov[28] superseded an earlier review by the same author.[68] In the high temperature region, Bodryakov[28] selected specific heat values which differed sinusoidally from the values selected here trending to 2.2% low at 1000 K to 5.8% high at 2400 K to 1.8% low at 3250 K. Although the specific heat curve selected by Bodryakov behaved more naturally than the presently selected curve, integration leads to enthalpy values which exceed smoothed experimental values by an average of 1290 J/mol at 3000 K and above. On these grounds, the selected values of JANAF (Chase[26]), Maglić[27] and Bodryakov[28] are considered as not being representative of the actual high temperature experimental enthalpy values.
7 Summary of Representative Equations
High temperature specific heat equations are given in Table 7, free energy equations are given in Table 8 and transition values involved with the free energy equations in Table 9.
8 Deviations from the Selected Values
Deviations of superconducting specific heat values are given in Table 10 and low temperature normal state specific heat in Table 11. Deviations of high temperature enthalpy and specific heat values in the solid range are given in Tables 12 and 13 respectively.
9 Thermodynamic Tables
Low temperature thermodynamic properties of the solid are given in Tables 14 and 15 and of the high temperature thermodynamic properties of the condensed phases in Table 16. Thermodynamic properties of the gas are given in Table 17 whilst the vapour pressure summary is given in Table 18.
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Arblaster, J.W. Thermodynamic Properties of Tantalum. J. Phase Equilib. Diffus. 39, 255–272 (2018). https://doi.org/10.1007/s11669-018-0627-2
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DOI: https://doi.org/10.1007/s11669-018-0627-2