*Johnson Matthey Technol. Rev.*, 2017, **61**, (2), 80

doi:10.1595/205651317x694461

## 液态铂族金属的密度和摩尔体积以及铱、铑、钌初期熔化曲线的选值

### 评估用于测定液态铂族金属密度的不同方法

建议使用给定方程式来表征温度给液态铂族金属的密度和摩尔体积所带来的差异，同时估计铱、銠、钌的熔化曲线中此前未知的初始斜率。

## Selected Values for the Densities and Molar Volumes of the Liquid Platinum Group Metals and of the Initial Melting Curves of Iridium, Rhodium and Ruthenium

### Assessing different determinations of the density of the liquid platinum group metals

- John W. Arblaster
- Droitwich, Worcestershire, UK
- Email: jwarblaster@yahoo.co.uk

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

Authors | Ref. | T_{m}, 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 d*D*/d*T* at the melting point from the quadratic fits to the density values (Equations (i) to (iv)):

where Δ*V * is the difference between the molar volumes of the solid and the liquid, Δ*H* is the enthalpy of fusion in J mol^{–1} at melting point *T*_{m}, K, and d*P*/d*T* is the initial slope of the melting curve in MPa K^{–1}. In the case of iridium, rhodium and ruthenium d*P*/d*T* 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 d*P*/d*T* 22.5 ± 1.3 MPa K^{–1} taken from an average of the d*T*/d*P* 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 Δ*V* = 0.482 ± 0.034 cm^{3} 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 Δ*V* at 0.582 cm^{3} mol^{–1} by Stankus and Khairulin (5) and 0.696 cm^{3} mol^{–1} by Pottlacher (2). The density thermal expansion at *b * = –0.96 cm^{3} 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 cm^{3} 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:

## 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 d*P*/d*T* 21.7 ± 2.2 MPa K^{–1}, equivalent to the d*T*/d*P* value of 46 K GPa^{–1} as determined by Errandonea (9) and assuming an accuracy of 10%. The resultant value is Δ*V* = 0.405 ± 0.045 cm^{3} 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 Δ*V* at 0.506 cm^{3} 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 cm^{3} mol^{–1} K^{–1} as determined by Paradis *et al.* (15) is confirmed by the value of –0.734 cm^{3} mol^{–1} K^{–1} obtained by Stankus and Tyagel’skii (3) and by the value –0.733 cm^{3} 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:

## 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 d*P*/d*T* 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 Δ*V* = 0.555 ± 0.191 cm^{3} 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 d*T*/d*P* 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 d*T*/d*P* is due almost entirely to the 2% accuracy assigned to the liquid density value and its effect on Δ*V* . 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 Δ*V* = 0.555 cm^{3} 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:

## 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 Δ*V* = 0.666 ± 0.197 cm^{3} 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 d*T*/d*P* 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 d*T* /d*P* is due almost entirely to the 2% accuracy assigned to the liquid density value and its effect on Δ*V*. The density equation of Ishikawa *et al.* (21) which covers the range 2300 to 3000 K is reproduced as Equation (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 Δ*V* = 0.532 ± 0.189 cm^{3} 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 d*P*/d*T* is 28.2 ± 10.1 MPa K^{–1} or the equivalent d*T*/d*P* 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 d*T*/d*P* 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)):

## 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)):

## 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 d*T*/d*P* 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.

##### Table II

^{a}Liquid density values are fitted to the equation: *D* (kg m^{–3}) = *a * – *b * (*T* – *T*_{m}) where *a * is the density at the melting point *T*_{m}, K, and *b * is the density thermal expansion. The temperature range approximates to the actual experimental range and therefore the density values mainly correspond to the undercooled region

##### Table III

^{a}Because liquid density values vary linearly with temperature then the reciprocal, molar volumes, can only be accurately represented by fitting to quadratic equations. The molar volume is given by *V*_{m} (cm^{3} mol^{–1}) = *A*_{r}/*D* = *c * + *d T* + *e T* ^{2} where *A*_{r} is the atomic weight and these equations are considered to be valid over the same temperature ranges as adopted for the liquid metals. The 2015 atomic weights were adopted (32)

##### Table IV

^{a}Values are based on the equations given in **Tables III** and **IV**. A conservative value for the accuracy of the density of platinum is assumed to be ±100 kg m^{–3} whilst all other density values are assumed to be accurate to 2%. Molar volumes are extended to three decimal places for interpolation purposes

##### Table V

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## The Author

John W. Arblaster is interested in the history of science and the evaluation of the thermodynamic and crystallographic properties of the elements. Now retired, he previously worked as a metallurgical chemist in a number of commercial laboratories and was involved in the analysis of a wide range of ferrous and non-ferrous alloys.