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Platinum Metals Rev., 2005, 49, (3), 141

doi:10.1595/147106705x54262

The Thermodynamic Properties of Platinum

REVISED DATA FOR THE LIQUID STATE AND VAPOUR PRESSURE

  • J. W. Arblaster
  • Coleshill Laboratories,
  • Gorsey Lane, Coleshill, West Midlands B46 1JU, U.K.
  • Email: jwarblaster@aol.com
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Article Synopsis

The thermodynamic properties of platinum were reviewed here by the author in 1994. However, the use of superior values for the enthalpy of liquid platinum have resulted in a major revision of the thermodynamic values for the liquid phase. One consequence of the revision is that the boiling point at one atmosphere pressure is altered from 4125 K, as calculated in the 1994 paper, to 4149 K. Previously accepted values for solid platinum are not altered, and there are only very minor changes to the gaseous phase values as a result of using a new atomic weight for platinum and the CODATA 2002 recommended values of the fundamental constants.

Platinum has a face centred cubic structure with a lattice parameter at 20ºC of 0.39236 nm and a density of 21.45 × 103 kg m–3 (1). In the present paper the variation of the thermodynamic values of the specific heat at constant pressure, enthalpy, entropy and Gibbs free energy with temperature have been revised in the condensed phases and gaseous phase at a 1 bar standard state pressure. The vapour pressure data was calculated from the selected heat of sublimation of 565 kJ mol–1 as shown in Table I and from the nett Gibbs free energy between the condensed and gaseous phases. Values for these properties are reassessed in the light of revised data.

Table I

Third Law Heats of Sublimation of Platinum

ReferenceTemperature range, KΔH°298.15, kJ mol–1Notes
Koch, Calvert, Thomas and Beall (13) 2032–2445 559.4 ± 1.1 a
Jones, Langmuir and Mackay (14) 1697–2034 564.8 ± 1.7 b
Dreger and Margrave (15) 1573–1785 566.5 ± 1.4 c
Hampson and Walker (16) 1918–2049 565.7 ± 0.5 d
Plante, Sessoms and Fitch (17) 1675–1977 564.4 ± 0.2 e
Selected value for the heat of sublimation of platinum at 298.15 K:
565 ± 2*

[i] a Weighted average of two data sets

b Temperatures corrected to ITS-90 from a laboratory temperature scale

c Two data points rejected by the authors

d Eight data points rejected by the authors

e Weighted average of eight data sets

* Following the practice of the National Institute of Standards and Technology (the former National Bureau of Standards), the accuracy quoted for the individual heats of sublimation are standard errors not standard deviations. Therefore the value quoted for the "accuracy" of the selected value tries to take into account not only the variations in the quoted values but also individual internal variations in the vapour pressure data sets. This practice is commonly used and represents a "real" accuracy rather than a mathematically derived accuracy which would tend to be lower and not realistic

Solid Phase

The thermodynamic properties of platinum were reviewed in this Journal in 1994 (2). The current revision leaves all values for the solid phase unaltered, with equations representing the variation with temperature of enthalpy and specific heat at constant pressure given in the box below. These equations have an overall accuracy of 0.3 per cent, equivalent to a standard deviation of ± 0.09 J mol–1 K–1 for the specific heat values. These equations were used to generate the thermodynamic properties of the solid phase shown in Table IIA.

Table IIA

High Temperature Data of Platinum in the Condensed Phase (solid, liquid)

T, Kp, J mol–1 K–1T – H°298.15, J mol–1T, J mol–1 K–1–(G°T – H°298.15)/T, J mol–1 K–1
298.15 (solid) 25.648 0 41.533 41.533
300 25.663 47 41.692 41.533
400 26.380 2125 49.176 42.549
500 26.986 5320 55.130 44.490
600 27.534 8046 60.099 46.688
700 28.049 10826 64.382 48.917
800 28.545 13655 68.160 51.091
900 29.036 16535 71.551 53.179
1000 29.531 19463 74.635 55.173
1100 30.040 22441 77.474 57.073
1200 30.575 25472 80.110 58.884
1300 31.144 28557 82.580 60.613
1400 31.757 31702 84.910 62.266
1500 32.422 34911 87.123 63.850
1600 33.150 38189 89.239 65.371
1700 33.949 41543 91.272 66.835
1800 34.829 44981 93.237 68.248
1900 35.799 48512 95.146 69.613
2000 36.869 52144 97.009 70.937
2041.3 (s) 37.314 53677 97.767 71.472
2041.3 (liquid) 38.993 75790 108.600 71.472
2100 38.993 78079 109.706 72.525
2200 38.993 81979 111.520 74.257
2300 38.993 85878 113.253 75.915
2400 38.993 89777 114.912 77.505
2500 38.993 93676 116.504 79.034
2600 38.993 97576 118.034 80.504
2700 38.993 101475 119.505 81.922
2800 38.993 105374 120.923 83.290
2900 38.993 109273 122.292 84.611
3000 38.993 113173 123.613 85.889
3100 38.993 117072 124.892 87.127
3200 38.993 120971 126.130 88.326
3300 38.993 124871 127.330 89.490
3400 38.993 128770 128.494 90.620
3500 38.993 132669 129.624 91.719
3600 38.993 136568 130.723 92.787
3700 38.993 140468 131.791 93.827
3800 38.993 144367 132.831 94.840
3900 38.993 148266 133.844 95.827
4000 38.993 152166 134.831 96.790
4100 38.993 156065 135.794 97.729
4200 (l) 38.993 159964 136.733 98.647

[i] s is solid, l is liquid

The values of enthalpy and entropy at 298.15 K are shown below.

Platinum at 298.15 KSolidGas
Enthalpy: H°298.15 – H°0, J mol-1 5694 6576.6
Entropy: S°298.15, J mol-1 K-1 41.53 192.409

Recent measurements of the enthalpy of platinum by Wilthan and colleagues (3, 4) using differential scanning calorimetry (at 473–1573 K) lead to values 3.6% higher falling to 1.7% higher than the selected values, while rapid pulse heating measurements by the same authors (at 1700–2040 K) lead to values of 1.1 to 1.4% lower.

Equations for the Enthalpy and Specific Heat at Constant Pressure of Solid Platinum
T – H°298.15 = 23.8992 T + 3.949695 × 10–3 T2 – 1.25821 × 10–6 T3 + 3.836275 × 10–10 T4 + 27697.5/T – 7539.23 J mol–1
C°p = 23.8992 + 7.89939 × 10–3 T – 3.77463 × 10–6 T2 + 1.53451 × 10–9 T3 – 27697.5/T2 J mol–1 K–1

Gaseous Phase

For the gaseous phase the introduction of a new atomic weight for platinum of 195.078 (5) and of the CODATA 2002 fundamental constants (6) have caused only two values (the free energy at 2800 K and the entropy at 3400 K) to be altered, but only by one digit in the last decimal place. Thermodynamic properties of the gaseous phase are shown in Table IIB and the values of enthalpy and entropy at 298.15 K are on the previous page.

Table IIB

High Temperature Data of Platinum in the Gaseous Phase (gas, 1 bar pressure)

T, Kp, J mol–1 K–1T – H°298.15, J mol–1T, J mol–1 K–1–(G°T – H°298.15)/T, J mol–1 K–1
298.15 25.531 0 192.409 192.409
300 25.577 47 192.567 192.410
400 27.023 2694 200.172 193.437
500 26.923 5400 206.210 195.410
600 26.191 8058 211.059 197.628
700 25.349 10635 215.032 199.840
800 24.591 13131 218.366 201.953
900 23.965 15557 221.225 203.939
1000 23.468 17928 223.723 205.795
1100 23.083 20255 225.941 207.528
1200 22.791 22548 227.937 209.147
1300 22.574 24815 229.752 210.663
1400 22.418 27065 231.419 212.087
1500 22.313 29301 232.962 213.428
1600 22.249 31529 234.399 214.694
1700 22.220 33752 235.747 215.893
1800 22.219 35973 237.017 217.032
1900 22.241 38196 238.219 218.116
2000 22.283 40422 239.361 219.150
2041.3 22.305 41343 239.816 219.563
2100 22.341 42653 240.449 220.138
2200 22.412 44891 241.490 221.085
2300 22.494 47136 242.488 221.994
2400 22.584 49390 243.447 222.868
2500 22.682 51653 244.371 223.710
2600 22.784 53926 245.263 224.522
2700 22.891 56210 246.125 225.306
2800 23.001 58505 246.959 226.064
2900 23.113 60810 247.768 226.799
3000 23.226 63127 248.554 227.511
3100 23.340 65456 249.317 228.202
3200 23.453 67795 250.060 228.874
3300 23.566 70146 250.783 229.527
3400 23.678 72508 251.488 230.162
3500 23.789 74882 252.176 230.782
3600 23.898 77266 252.848 231.385
3700 24.005 79661 253.504 231.974
3800 24.111 82067 254.146 232.549
3900 24.215 84483 254.774 233.111
4000 24.318 86910 255.388 233.660
4100 24.418 89347 255.990 234.198
4200 24.517 91794 256.579 234.724

Liquid Phase

In 1994, when the original paper was published (2), the melting point of platinum, 2041.3 K, was only tentatively assigned to the ITS-90 temperature scale. It was fully accepted later (7).

However, selected values used in that paper, in the liquid region, were based on the enthalpy values of Chaudhuri et al. (8) which were supposedly the measurements made by David Bonnell who was working on his Ph.D. thesis at Rice University. Bonnell's Ph.D. thesis was published two years later (9) and it was assumed that this was because he was completing work on other metals and that, as is the usual practice, the values in the earlier paper were definitive. However, private communication with Bonnell has indicated that this was not the case. In fact, the thesis contains the fully corrected values. The values given in the paper published two years earlier were only preliminary and should not have been given the attention they received.

Therefore, in now selecting the values given in Bonnell's thesis, his temperature values in the range 2205 to 2650 K are accepted for this review; while his enthalpy values have been adjusted by 195.08/195.09 to ensure that the atomic weight is the same as that used previously for the solid phase (2). These measurements can then be represented by the following equation which has an overall accuracy of 1.2%, equivalent to a standard deviation of ± 1190 J mol–1:

T – H°298.15 = 38.9928 T – 3805.63 J mol–1

Thus, derived values are:

  • Specific heat (Cp): 39.0 ± 2.2 J mol–1 K–1

  • Heat of fusion: 22.11 ± 0.94 kJ mol–1 and

  • Entropy of fusion: 10.83 ± 0.46 J mol-1 K-1.

The difference between the above three values and those tabulated for the liquid phase in Table IIA is due to the values in the Table being taken beyond their true accuracy for interpolation purposes.

Other measurements made on platinum in the liquid region used the rapid pulse heating technique, but the different values obtained show marked differences, and currently this method does not appear to be as accurate as the levitation calorimetry method used by Bonnell.

For instance, Gather, Shaner and Hodgson (10) (at 2041–8000 K) obtained a value for the specific heat of 49 J mol–1 K–1 and for the heat of fusion of 27 ± 6 kJ mol–1, while Lebedev, Savvatimskii and Smirnov (11) obtained a heat of fusion of 25 kJ mol–1.

The two most recent sets of measurements: by Hixson and Winkler (12) (at 2041–5100 K) lead to a specific heat of 41.35 J mol–1 K–1 and a heat of fusion of 24.2 kJ mol–1, and the measurements of Wilthan and colleagues (5, 6) (at 2045–2830 K) lead to a specific heat of 36.5 J mol–1 K–1 and a heat of fusion of 21.8 kJ mol–1. On average these latter two sets of results show satisfactory agreement with values selected here.

Vapour Pressure

Only the measurements of Koch et al. (13) on liquid platinum are affected by the revision, and even in this case the derived heat of sublimation is only lowered by 0.1 kJ mol1, see Table I. However, these measurements of Koch et al. are still discrepant when compared to the other four values obtained on solid platinum as shown in Table I and have again been rejected. In selecting the heat of sublimation most weight is given to the measurements of Hampson and Walker (16) and those of Plante, Sessoms and Fitch (17).

Table III

Vapour Pressure Data of Platinum

T, KP, barΔG°, J mol–1ΔH°, J mol–1P, barT, K
298.15 (solid) 7.89 × 10–92 520016 565000 10–12 1489
400 1.26 × 10–66 506645 565043 10–11 1569
500 7.23 × 10–52 489540 565080 10–10 1659
600 4.98 × 10–42 474436 565012 10–9 1759
700 5.29 × 10–35 459354 564809 10–8 1872
800 9.77 × 10–30 444310 564776 10–7 2002
900 1.21 × 10–25 429316 564022 10–6 2156
1000 2.27 × 10–22 414378 563465 10–5 2339
1100 1.07 × 10–19 399500 562814 10–4 2558
1200 1.80 × 10–17 384684 562076 10–3 2824
1300 1.37 × 10–15 369935 561258 10–2 3155
1400 5.57 × 10–14 355251 560363 10–1 3580
1500 1.38 × 10–12 340633 559390 1 4146
1600 2.26 × 10–11 326083 558340 NBP 4149
1700 2.67 × 10–10 311601 557209
1800 2.38 × 10–9 297189 555992
1900 1.68 × 10–8 282844 554684 NBP: normal boiling point at a pressure of one atmosphere (1.01325 bar)
2000 9.68 × 10–8 268574 553278
2041.3 (s) 1.90 × 10–7 262702 552666
2041.3 (liquid) 1.90 × 10–7 262702 530553
2100 4.54 × 10–7 255013 529574 ΔH°0 = 564.117 ± 2.000 kJ mol–1
2200 1.80 × 10–6 241978 527912
2300 6.29 × 10–6 229018 526258
2400 1.98 × 10–5 216130 524613
2500 5.65 × 10–5 203310 522977
2600 1.49 × 10–4 190555 521351
2700 3.62 × 10–4 177863 519735
2800 8.27 × 10–4 165230 518131
2900 1.78 × 10–3 152655 516537
3000 3.63 × 10–3 140134 514955
3100 7.06 × 10–3 127666 513384
3200 1.31 × 10–2 115249 511824
3300 2.35 × 10–2 102880 510276
3400 4.06 × 10–2 90557 508739
3500 6.79 × 10–2 78280 507213
3600 0.110 66047 505698
3700 0.174 53855 501494
3800 0.267 41703 502700
3900 0.402 29591 501217
4000 0.591 17517 499745
4100 0.852 5479 498282
4200 (l) 1.205 –6522 496830

[i] s is solid, l is liquid

Vapour Pressure Equations
Solid: 1200–2041.3 K: ln(P, bar) = 28.3308 – 1.29944 ln(T) – 69207.9/T
Liquid: 2041.3–4200 K: ln(P, bar) = 32.1390 – 1.89944 ln(T) – 67647.6/T

Previous accepted values for the vapour pressure of the solid phase and revised values for the liquid phase are given in Table III.

Vapour pressure equations for both the liquid and the gas are given above. For the solid the values are given over a practical range from 1200 K to the melting point of platinum, while although the values for the liquid are given over a much larger temperature range from the melting point to 4200 K the derived normal boiling point is only 0.2 K higher than that obtained from a proper thermodynamic treatment.

Free Energy Equations

The free energy equations, see below, are derived from those given above for the solid and liquid phases. It is a requirement that the two equations given must be equal at the melting point.

Free Energy Equations
Solid: G°T – H°298.15 = 120.8910 T – 3.949695 × 10–3 T 2 + 6.29105 × 10–7 T 3 – 1.278758 × 10–10 T 4 + 13848.75/T – 23.8992 T ln (T) – 7539.23 J mol–1
Liquid: G°T – H°298.15 = 227.5700 T – 38.9928 T ln (T) – 3805.63 J mol–1

Conclusions

The thermodynamic properties of platinum have been revised by introducing superior values for the enthalpy of the liquid. The vapour pressure curve for the liquid is also reassessed as shown in Table III. Sets of revised values for specific heat at constant pressure, enthalpy, entropy, Gibbs' free energy and vapour pressure are presented.

Glossary of Thermodynamic Terms
T Temperature
R Gas constant
Cop Specific heat at constant pressure
Ho Enthalpy = ∫Cop(T).dT between temperatures T1 and T2
So Entropy = ∫(Cop(T)/T).dT between temperatures T1 and T2
Go Gibbs free energy = Ho – TSo
– Go/T Gibbs free energy/temperature = So – Ho/T
ΔHo298.15 Heat of sublimation at 298.15 K
ΔHo Enthalpy of sublimation at (solid) or evaporation (liquid)
ΔGo Free energy of sublimation at (solid) or evaporation (liquid)
P Vapour pressure in pascals (Pa) or the equivalent bars (105 Pa). The terms torr (mm of mercury pressure) and atmosphere are now obsolete although the normal boiling point is still quoted at one atmosphere pressure (1.01325 bar)
Analytical Expressions
Solid
Cop = a + bT + cT2 + dT3 + e/T2
HoT – Ho298.15 = aT + (b/2)T2 + (c/3)T3 + (d/4)T4 – e/T + f
So = a ln(T) + bT + (c/2)T2 + (d/3)T3 – (e/2)/T2 + g
– (GoT – Ho298.15)/T = a ln(T) + g – a + (b/2)T + (c/6)T2 + (d/12)T3 + (e/2)/T2 – f/T
GoT – Ho298.15 = – a ln(T) + T(a – g) – (b/2)T2 – (c/6)T3 – (d/12)T4 + (e/2)/T + f
"f" is evaluated at 298.15 K where HoT – Ho298.15 = 0; "g" is evaluated at 298.15 K by fixing the value of So298.15
Liquid
HoT – Ho298.15 = aT + f
So = a ln(T) + g
– (GoT – Ho298.15)/T = a ln(T) + g – a – f/T
GoT – Ho298.15 = – a ln(T) + T(a – g) + f
"f" and "g" are evaluated from the values of enthalpy and entropy for the liquid at the melting point
Calculation of the Thermodynamic Properties of an Ideal Monatomic Gas

The thermodynamic properties of platinum gas given in Table IIB were calculated using the method of Kolsky, Gilmer and Gilles (18) outlined below. All values given correspond to the 2002 CODATA fundamental constants (6) and to a standard state pressure of one bar.

Velocity of light, c = 299792458 m s–1 exactly

Avogadro constant, NA = (6.0221415 ± 0.0000010) × 1023 mol–1

Planck constant, h = (6.6260693 ± 0.0000011) × 10–34 J s

Gas constant, R = 8.314472 ± 0.000015 J mol–1 K–1

Second radiation constant, c2 = 0.014387752 ± 0.000000025 m K (metre Kelvin)

Atomic weight, Ar

Energy level = νi cm–1; Quantum number of energy level, Ji

α = c2 = 1.4387752 cm K

K1 = Rα = 11.962656 J cm mol–1

K2 = 2.5R = 20.786180 J mol–1 K–1

K3 = Rα2 = 17.211573 J cm2 K mol–1

K4 = 1.5R = 12.471708 J mol–1 K–1

K5 = S0 = – 9.5758165 J mol–1 K–1

K6 = K5 – 2.5R = – 30.3619965 J mol–1 K–1

S0, the Sackur-Tetrode constant, is derived from: S0 = R[2.5 + ln 4.980463969 x 10–9 R5/2/h3 NA 4)], where the numerical value in the equation is equal to (2π/1000)3/2/105

Partition Functions

Q = ∑(2Ji + 1)e–ανi/T

Q1 = ∑(2Ji + 1)νi e–ανi/T

Q2 = ∑(2Ji + 1)νi2 e–ανi/T

Thermodynamic Equations

Values are initially referred to 0 K:

Cop = (K3/T2)(Q2/Q1 – (Q1/Q)2) + K2 J mol–1 K–1

Hop – Ho0 = K1(Q1/Q) + K2 T J mol–1

So = (K1/T)(Q1/Q) + R ln(Q) + K4 ln(Ar) + K2 ln(T) + K5 J mol–1 –1

– (GoT – Ho0)/T = R ln(Q) + K4 ln(Ar) 2 ln(T) + K6 J mol–1 K–1 = So – (HoT o0)/T J mol–1 K–1

HoT – Ho298.15 = (HoT o0) – (Ho298.15– Ho0) J mol–1

– (GoT – Ho298.15)/T = – (GoT – Ho0)/T + (Ho298.15 – Ho0)/T J mol–1 K–1

Vapour Pressure

ΔHo = ΔHo298.15 + δ(HoT – Ho298.15)

ΔGo = ΔHo298.15 – Tδ – (GoT – Ho298.15)/T

ln(P) = – ΔGo/RT

δ(HoT – Ho298.15) = (HoT – Ho298.15) (gas) – (HoT – Ho298.15) (solid, liquid)

δ – (GoT – Ho298.15)/T = – (GoT – Ho298.15)/T (gas) – – (GoT – Ho298.15)/T (solid, liquid)

Calculation of the Heat of Sublimation from Vapour Pressure Data (as relating to Table I )

Calculated values are the Second Law heat of sublimation, ΔH o 298.15 (II), and the Third Law heat of sublimation, ΔH o 298.15 (III). Significant differences between these two evaluations may indicate that the vapour pressure measurements are erroneous. The selected heat of sublimation is calculated from the Third Law values.

Third Law Heat of Sublimation

Each data point is evaluated separately and the selected value is based on averaging the derived heats of sublimation:

ΔH o 298.15 (III) = T[δ – (G o T – H o 298.15 )/T – R ln(P, bar)]

Revised Second Law Heat of Sublimation

All data points are fitted to the following equation:

δ – (G o T – H o 298.15 )/T – R ln(P, bar) = B + A/T where A = ΔH o 298.15 (II) and B = entropy drift = δS o 298.15 (III) – δS o 298.15 (II)

Ideally B should be zero but if it is an unacceptably large number then this may again indicate that the vapour pressure measurements are erroneous.

Traditional Second Law Heat of Sublimation

All data points are fitted to the Clausius-Clapeyron equation:

ln(P, bar) = B + A/T

ΔH o 298.15 (II) = – δ(H o T – H o 298.15 )) – R A

Temperature is either the average experimental value or the average value obtained from reciprocal temperatures, but there is no formal definition.

"Pseudo" Third Law Heat of Sublimation

For vapour pressure measurements given only in the form of an equation, usually the Clausius-Clapeyron equation, Third Law values are calculated at the extreme ends of the temperature range and averaged. Where possible the assigned accuracy is that given by the authors who generally carried out a proper Third Law evaluation using previously selected thermodynamic values but reported the results only in the form of the above equation.

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References

  1.  J. W. Arblaster, Platinum Metals Rev., 1997, 41, (1), 12
  2.  J. W. Arblaster, Platinum Metals Rev., 1994, 38, (3), 119
  3.  B. Wilthan, C. Cagran, C. Brunner and G. Pottlacher, Thermochim. Acta, 2004, 415, 47
  4.  B. Wilthan, C. Cagran and G. Pottlacher, Int. J. Thermophys., 2004, 25, 1519
  5.  R. D. Loss, Pure Appl. Chem., 2003, 75, 1107
  6.  P. J. Mohr and B. N. Taylor, Rev. Mod. Phys., 2005, 77, 1
  7.   R. E. Bedford, G. Bonnier, H. Maas and F. Pavese, Metrologia, 1996, 33, 133
  8.  A. K. Chaudhuri, D. W. Bonnell, L. A. Ford and J. L. Margrave, High Temp. Sci., 1970, 2, 203
  9.  D. W. Bonnell, "Property Measurements at High Temp., Levitation Calorimetry Studies of Liquid Metals", Ph.D. Thesis, Rice Univ., Houston, Texas, 1972
  10.  G. K. Gathers, J. W. Shaner and W. M. Hodgson, High Temp.-High Pressures, 1979, 11, 529
  11.  S. V. Lebedev, A. I. Savvatimskii and Yu. B. Smirnov, Teplofiz. Vys. Temp., 1971, 9, 635; High Temp., 1971, 9, 578
  12.  R. S. Hixson and M. A. Winkler, Int. J. Thermophys., 1993, 14, 409
  13.  R. K. Koch, E. D. Calvert, C. R. Thomas and R. A. Beall, U.S. Bur. Mines Rep. Invest. 7271, July, 1969
  14.  H. A. Jones, I. L. Langmuir and G. M. Mackay, Phys. Rev., 1927, 30, 201
  15.  L. H. Dreger and J. L. Margrave, J. Phys. Chem., 1960, 64, 1323
  16.  R. F. Hampson and R. F. Walker, J. Res. Nat. Bur. Stand., 1961, 65A, 289
  17.  E. R. Plante, A. B. Sessoms and K. R. Fitch, J. Res. Nat. Bur. Stand., 1970, 74A, 647
  18.  H. G. Kolsky, R. M. Gilmer and P. W. Gilles, United States Atomic Energy Report, LA 2110, March, 1957

The Author

John W. Arblaster is Chief Chemist working in metallurgical analysis on a wide range of ferrous and non-ferrous alloys for standards in chemical analysis at Coleshill Laboratories, in the West Midlands of England. He is interested in the history of science and in the evaluation of the thermodynamic and crystallographic properties of the elements.

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