The preliminary results on heat capacity of tantalum up to 3000 K were reported in an earlier publication [3]. In this study the measurements were extended to 3200 K by performing one experiment each on Ta-3 and Ta-4, and averaging the results. Hemispherical total emittance needed to correct heat capacity above 3000 K was obtained from the extrapolation of the experimental results at lower temperatures. The results of measurements above 3000 K are given in table 2. At 3000 K the heat capacity for this extended temperature range is approximately 0.1 percent lower than the value given by eq (9).

5.4. Electrical Resistivity

The electrical resistivity of the Ta-1 specimen was determined from the same experiments that were used

to calculate the heat capacity. A second degree poly nomial function for each heating series was obtaine by least squares approximation of results from ind vidual experiments. The standard deviation of th points from the function for the first and second heatin series are 0.03 and 0.02 percent, respectively.

In contrast to the heat capacity results, the electrica resistivity showed a small but significant differen between the two heating series; the results of th second series being lower than those of the first. Th combined data from both heating series were fitte to a second degree polynomial function. Figure 7 show the deviations of the experimental results from th smooth function for the combined series. The figur also shows the deviation of the function for each in dividual heating series from the electrical resistivit function for the combined series. The average diffe ence between the functions for the first and secon heating series is approximately 0.15 percent. Th function for the combined series that is valid for th temperature range 1900 to 3000 K is:

p=3.671+ 4.292 × 10-27-2.677 × 10-672 (10

where T is in K and p in 10-80 m. Electrical resistivit up to 3000 K computed using the above equation i given in table 2.

The difference between the results on Ta-1 an Ta-2 is presented graphically in figure 6. In contras to heat capacity, electrical resistivity did not converg as Ta-2 was exposed to high temperatures.

The results of experiments on Ta-3 and Ta-4 wer used to compute electrical resistivity above 3000 M These are included in table 2. At 3000 K electrica resistivity obtained from Ta-3 and Ta-4 was approx mately 0.3 percent lower than the value given by eq(10)

DEVIATION, PERCENT

-0.1

0.1

Numbers listed under imprecision were obtained from a least squares analysis of experimental results. Numbers listed under inaccuracy were estimated considering the contribution of various items that introduce random and systematic errors in the pertinent quantities. These items are listed below:

(a) In temperature measurements: pyrometer reproducibility, scattered light correction, light source alinement, radiation standard lamp, blackbody quality, specimen temperature uniformity, magnetic fields.

(b) In electrical measurements: skin effect, inductive effects, thermoelectric effects.

(c) In interpretation of results: specimen evaporation, thermionic emission, time synchronization, measurements of length and weight.

Details regarding the estimates of errors and their combination are given in another publication [1]. Specific items in the error analysis were recomputed whenever the present conditions differed from those in the earlier publication.

7. Discussion

The heat capacity and electrical resistivity results of this work are compared graphically with those in the literature in figures 8 and 9, respectively. Numerical comparisons are given in tables 4 and 5. It may be seen that present results agree favorably with all others at 2000 K and also at higher temperatures, with the exception of heat capacity results of Hoch and Johnston [8]. Estimates of errors in papers cited lead to an estimate of inaccuracies in previously reported heat capacity and electrical resistivity of approxi

FIGURE 11. Normal spectral emittance of tantalum at λ= 650 nm reported in the literature.

10-3) were obtained from data on heat capacity at ow and moderate temperatures (at 250 and 1000 K) given by Hultgren et al. [11].

Using eq (11) and the heat capacity results of this work, the quantity Ac was computed for temperatures bove 1900 K. The results are tabulated in table 7. The -stimated uncertainty in the computed Ac may be as igh as 1 J mol-1K-1. This was obtained from the combined uncertainties in the coefficients in eq (11) nd the measured heat capacities.

where

NAAE

=

Cvac

e-Eg/kT

kT2

NA Avogadro's number

=

k= Boltzmann constant

Ef vacancy formation energy

(12)

A = constant which is obtained from vacancy concentration at the melting point.

If one assumes that vacancy formation energy is approximately proportional to the melting point and considers the value of 3.3 eV reported by Schultz [9] for tungsten, one obtains 2.9±0.5 eV for the va cancy formation energy for tantalum. There are no accurate measurements on tantalum related to vacancy concentrations. Results of quenching experiments on various refractory elements [9, 10] have indicated that vacancy concentrations are probably in the range 0.01 to 0.1 percent at their melting points. Estimates corresponding to a vacancy concentration of 0.1 percent at the melting point and a vacancy formation energy of 2.9 eV are given in table 7. The results indicate that vacancy contribution is small, less than 0.7 J mol-1K-1 (upper limit) at 3200 K, and does not account for high heat capacity values. A

possible contribution of higher order terms in the electronic heat capacity may partially account for high values of heat capacity at high temperatures.

If the entire deviation of measured heat capacity from the sum of 3R and the linear term at high temperatures is represented by an expression similar to eq (12), one could obtain after rearrangement

This equation indicates that a plot of the left side. versus 1/7 should yield a straight line with slope equal to S. From the data on tantalum in the range 1900 to 3200 K, a straight line with a standard deviation of 0.8 percent was obtained. However, the parameters obtained through this fit do not seem to have any physical significance. As a crude analogy to vacancy concentration, the computations yielded a value of 1.4 eV for energy and 4.2 percent for concentration at the melting point. Both of these values seem to be unrealistic for tantalum.

In order to give a simple expression for the heat capacity of tantalum over a wide temperature range, an empirical term in 75 for the quantity Ac in eq (11) was substituted. The coefficient of this term was obtained from the results of the present work in conjunction with the values given by Hultgren et al. [11] at temperatures below 1000 K. Then, eq (11) for the range 300 to 3100 K becomes

what lower than the generally accepted value of 247 K [28]. Such a deviation may be expected since in the above analysis only data above 250 K were considered while determinations reported in the literature were based on more elaborate treatment of lower temperature data.

It was interesting to note that the difference in heat capacity between Ta-1 and Ta-2 was reduced from approximately 1 percent at 1900 K to 0.2 percent at 3000 K. The convergence of the results as Ta-2 was exposed to high temperatures indicates the difference in the initial states of the two specimens: Ta-1 was preheated while Ta-2 was used as received from the manufacturer prior to the start of the measurements. There was a small, but significant difference in electrical resistivity between the two heating series of Ta-1. The second series results, which were lower than those of the first by approximately 0.15 percent indicate that the specimen had undergone additiona annealing during its exposure to high temperatures

Unlike most metallic elements, the electrical resis tivity of tantalum, in the range of present measure ments, showed a negative departure from linearity i the curve of electrical resistivity against temperature A small Fermi energy is believed to be responsib for some of this negative departure [27].

The experimental results reported in this paper hav further substantiated the feasibility of accurate meas urement of heat capacity and electrical resistivity electrical conductors above approximately 2000 K b a pulse method of millisecond resolution. The resul also indicated that under proper surface and environ mental conditions the technique allows the measure ment of hemispherical total and normal spectr emittances.

The authors extend their appreciation to M. S. Mor for his contribution in connection with electronic strumentation, which is a vital part of the enti measurement system.