Definitive equations are suggested to represent the variation with temperature of the densities and molar volumes of the liquid platinum group metals whilst the previously unknown initial slopes of the melting curves for iridium, rhodium and ruthenium are estimated.

1. Introduction

Paradis et al. (1) summarised determinations of the densities of the liquid platinum group metals but a number of important determinations were not included. These are given in Table I in which values are presented as a – b (T – T_m) where a is the liquid density at the melting point T_m and b represents the density thermal expansion. Unfortunately, in the case of the measurements of Pottlacher (2) volume ratios were considered to vary linearly with temperature so that derived density values could only be fitted to quadratic equations (Equations (i)–(iv)), whereas a very large number of determinations of the densities of medium and high melting point metals and alloys clearly indicate that liquid density values vary linearly with temperature and this would therefore limit the usefulness of the values of Pottlacher (2). Paradis et al. (1) did not suggest definitive values for the densities of the metals but would undoubtedly have given preference to their own electrostatic levitation determinations which are considered to be accurate to about 2%. The problem with the selection was that the differences between various determinations were significant to a certain extent making an objective evaluation difficult. One alternative approach is to indirectly obtain the liquid density at the melting point from the value for the solid density by use of the Clausius-Clapeyron equation (Equation (v)):

Table I

Additional Determinations of the Densities of Liquid Platinum Group Metals

Authors	Ref.	Tm, K	a	b
Iridium
Pottlacher	2	2719	19,722	2.049*
Palladium
Pottlacher	2	1828	10,690	0.733*
Stankus and Tyagel’skii	3	1827	10,631	0.734
Popel et al.	4	1828	10,605	1.056
Platinum
Pottlacher	2	2042	18,968	1.170*
Stankus and Khairulin	5	2042	18,932	1.168
Rhodium
Pottlacher	2	2236	11,004	1.022*

*Initial slope dD/dT at the melting point from the quadratic fits to the density values (Equations (i) to (iv)):

 

(i)

 

(ii)

 

(iii)

 

(iv)

 

(v)

where ΔV is the difference between the molar volumes of the solid and the liquid, ΔH is the enthalpy of fusion in J mol^–1 at melting point T_m, K, and dP/dT is the initial slope of the melting curve in MPa K^–1. In the case of iridium, rhodium and ruthenium dP/dT is unknown so that Equation (v) can be reversed to estimate these values. In applying Equation (v) the densities of the solids at the melting point were all assumed to have an accuracy of ±20 kg m^–3 which is roughly twice the room temperature uncertainty and therefore equivalent to a 95% confidence level.

2. Platinum

Selected values for the fit to Equation (v) are melting point 2041.3 ± 0.4 K (6), enthalpy of fusion 22,113 ± 940 J mol^–1 (7) and slope of the melting curve dP/dT 22.5 ± 1.3 MPa K^–1 taken from an average of the dT/dP values of 42 K GPa^–1 as determined by Mitra et al. (8) and 47 K GPa^–1 as determined by Errandonea (9). The resultant value is ΔV = 0.482 ± 0.034 cm³ mol^–1 which when combined with the bulk density of the solid as 20,173 ± 20 kg m^–3 (10, 11) leads to a density of the liquid at the melting point of 19,215 ± 67 kg m^–3. This is in extraordinary agreement with the value of 19,200 ± 380 kg m^–3 determined by Ishikawa et al. (12). Other density determinations such as that of Stankus and Khairulin (5) at 18,932 ± 90 kg m^–3 and that of Pottlacher (2) at 18,968 kg m^–3 are notably lower but agree with other determinations summarised by Paradis et al. (1) suggesting that a possible value for the density could be 18.9 ± 0.1 kg m^–3 and that the determination of Ishikawa et al. (12) would then be an outlying value. However, the indirect density value obtained from the Clausius-Clapeyron equation (Equation (v)) clearly confirms the determination of Ishikawa et al. (12) as being the most likely value. The significant difference resulting in the lower density values can be traced to much larger values determined for ΔV at 0.582 cm³ mol^–1 by Stankus and Khairulin (5) and 0.696 cm³ mol^–1 by Pottlacher (2). The density thermal expansion at b = –0.96 cm³ mol^–1 K^–1 as determined by Ishikawa et al. (12) is much lower than previous values but is in reasonable agreement with the value of –1.168 ± 0.062 cm³ mol^–1 K^–1 as determined by Stankus and Khairulin (5). Therefore, a suggested equation (Equation (vi)) to represent the density of liquid platinum over the range from 1700 K to 2200 K with an accuracy of about 0.5% would be:

(vi)

3. Palladium

Selected values for the fit to Equation (v) are melting point 1828.0 ± 0.1 K (6), enthalpy of fusion 16,080 ± 740 J mol^–1 (13) and slope of the melting curve dP/dT 21.7 ± 2.2 MPa K^–1, equivalent to the dT/dP value of 46 K GPa^–1 as determined by Errandonea (9) and assuming an accuracy of 10%. The resultant value is ΔV = 0.405 ± 0.045 cm³ mol^–1 and when combined with the density of the solid of 11,179 ± 20 kg m^–3 (14) leads to a density of the liquid of 10,723 ± 50 kg m^–3 which is higher than any of the experimental values but is encompassed within the accuracy of the determination of Paradis et al. (15) at 10,660 ± 210 kg m^–3 and the value of Pottlacher (2) at 10,690 kg m^–3. Two further recent determinations are notably lower with Popel et al. (4) obtaining the value 10,605 kg m^–3 and Stankus and Tyagel’skii (3) obtaining 10,631 ± 45 kg m^–3, although in the case of the latter the value of ΔV at 0.506 cm³ mol^–1 is again much larger than the value obtained using Equation (v). It is known that the values of the enthalpy of fusion and the density of the solid at the melting point are tentative so that the value obtained from Equation (v) may not be fully representative. It is also noted that the average of the four sets of measurements considered above is very close to the value of Paradis et al. (15) and therefore this value is selected since it does include the Equation (v) value in its uncertainty. The density thermal expansion coefficient of b = –0.77 cm³ mol^–1 K^–1 as determined by Paradis et al. (15) is confirmed by the value of –0.734 cm³ mol^–1 K^–1 obtained by Stankus and Tyagel’skii (3) and by the value –0.733 cm³ mol^–1 K^–1 at the melting point calculated from the quadratic fit to the measurements of Pottlacher (2). Therefore, the equation (Equation (vii)) given by Paradis et al. (15) is considered as being representative for palladium over the temperature range 1600 to 1900 K when consideration is given to its 2% accuracy:

(vii)

4. Rhodium

Strong and Bundy (16) determined an initial slope of the melting curve of 62 K GPa^–1 but the value determined for platinum at the same time, 72 K GPa^–1, far exceeds the more recent determinations given above. Therefore, it is assumed that the slope dP/dT is poorly known and can be calculated by reversing Equation (v) with a melting point of 2236 ± 3 K (6), an enthalpy of fusion of 27,295 ± 850 J mol^–1 (17) and a value of ΔV = 0.555 ± 0.191 cm³ mol^–1 based on the density of the solid at the melting point at 11,491 ± 20 kg m^–3 (18) and for the liquid at 10,820 ± 220 kg m^–3 (19). The derived melting curve pressure is 22.0 ± 7.6 MPa K^–1 or the equivalent dT/dP value of 45 ± 16 K GPa^–1 which agrees closely with the values obtained for both platinum and palladium. The relatively poor accuracy assigned to dT/dP is due almost entirely to the 2% accuracy assigned to the liquid density value and its effect on ΔV . The density equation given by Paradis et al. (19) over the range 1820 to 2250 K has been repeated as Equation (viii) to remove the ambiguity created by Paradis et al. (1) who included two different melting point values. The quadratic fit to the density values of Pottlacher (2) (2236 to 3500 K) leads to a value of 11,004 kg m^–3 at the melting point which is encompassed within the accuracy assigned to the measurements of Paradis et al. (19). Perhaps coincidentally the volume ratios of Pottlacher (2) also lead to ΔV = 0.555 cm³ mol^–1 at the melting point although the value obtained of b = –1.022 kg m^–3 K^–1 is notably higher than the value given by Paradis et al. (19) in Equation (viii) below:

(viii)

5. Iridium

The melting point slope is unknown and was also derived by reversing Equation (v). Initially Ishikawa et al. (20) reported that their group (21) obtained a liquid density value of 19,870 kg m^–3 at the melting point although the actual published value had been reduced to 19,500 kg m^–3. For the reverse of Equation (v) input values are melting point 2719 ± 4 K (6), enthalpy of fusion 41,335 ± 1128 J mol^–1 (22) and ΔV = 0.666 ± 0.197 cm³ mol^–1 based on the density of the solid at the melting point at 20,913 ± 20 kg m^–3 (23) and for the liquid at the melting point 19,500 ± 390 kg m^–3 (21). The derived melting curve pressure is 22.8 ± 6.8 MPa K^–1 or the equivalent dT/dP value of 44 ± 13 K GPa^–1 which again agrees closely with the values obtained for both platinum and palladium. The relatively poor accuracy assigned to dT /dP is due almost entirely to the 2% accuracy assigned to the liquid density value and its effect on ΔV. The density equation of Ishikawa et al. (21) which covers the range 2300 to 3000 K is reproduced as Equation (ix):

(ix)

Measurements given by Pottlacher (2) were ambiguous since the baseline solid density was given as a value for commercial purity iridium at 22,420 kg m^–3 rather than the X-ray value of 22,560 kg m^–3 (23). Using the assigned commercial value, the liquid density value derived from the quadratic fit at 19,722 kg m^–3 is encompassed by the accuracy assigned to the measurement of Ishikawa et al. (20). However, the derived value at the initial slope of the melting curve derived from the quadratic fit at b = –2.049 kg m^–3 K^–1 differs considerably from the value given in Equation (ix).

6. Ruthenium

The only precision liquid density measurements are those of Paradis et al. (24) over the range 2225 to 2775 K. Again the slope of the melting curve was unknown and was derived by reversing Equation (v) using values of melting point 2606 ± 10 K (6), enthalpy of fusion 39,038 ± 1400 J mol^–1 (25) and ΔV = 0.532 ± 0.189 cm³ mol^–1 based on the density of the solid at the melting point 11,396 ± 20 kg m^–3 (26) and for the liquid 10,751 ± 210 kg m^–3 (24). The derived melting curve pressure dP/dT is 28.2 ± 10.1 MPa K^–1 or the equivalent dT/dP value of 36 ± 13 K GPa^–1 which is lower, but still within the accuracy limits, measured or derived for the face-centred cubic platinum group metals. For comparison theoretical dT/dP values for osmium vary between 40.4 K GPa^–1 (27) and 49.5 K GPa^–1 (28) in agreement with the face-centred cubic values. Paradis et al. (24) assumed a melting point of 2607 K for ruthenium rather than the International Temperature Scale (ITS-90) value of 2606 K. The published density equation has therefore been adjusted to correspond to the corrected melting point (Equation (x)):

(x)

7. Osmium

Actual density measurements of solid osmium extend only to 1300 K (29) and therefore estimating possible values above this temperature is speculative. Paradis and Ishikawa (30) measured the liquid density over the range 2670 to 3380 K and assumed that measurements were both undercooled and in equilibrium by taking the melting point to be 3306 K. However, this literature value was obtained on osmium metal of only commercial purity and is considered to be far too low. Arblaster (31) suggested that the true melting point of pure osmium was likely to be in the order of 3400 ± 50 K and the published equation was revised to correspond to this estimated melting point. However, this correction is only formal so that all melting points conform to selected values and the actual derived density values correspond to the experimentally determined values (Equation (xi)):

(xi)

8. Conclusions

An evaluation of different determinations of the density of the liquid platinum group metals concludes that the determinations using the electrostatic levitation method are possibly the most reliable and derived equations from this method are given in Table II, with slight modifications for the density of liquid platinum and the melting points of ruthenium and osmium. Derived molar volume equations are given in Table III whilst Table IV gives the values of density and molar volume at the melting points. Derived densities and molar volumes at other temperatures are given in Table V.

The initial slopes of the melting curves for ruthenium, rhodium and iridium were at first considered to be unknown and were derived using the Clausius-Clapeyron equation (Equation (v)). This showed that the initial slopes derived for rhodium and iridium were very close to the actual experimental values of palladium and platinum suggesting a common value for dT/dP of about 45 K GPa^–1 for the face-centred cubic platinum group metals. The value of 36 K GPa^–1 obtained for ruthenium and estimates of 40 to 50 K GPa^–1 for osmium suggest that a possible common value for the hexagonal close-packed platinum group metals is less certain.