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Canadian Metallurgical Quarterly, Vol. 23, No.3, pp. 309-313, 1984

Printed in Canada0008-4433/85 $3.00 + .00

Canadian Institute of Mining and Metallurgy

Pergamon Press Ltd.

/VAPOUR PRESSURE EQUATIONS

FOR THE METALLIC ELEMENTS: 298-2500K

C. B. ALCOCK, V. P. ITKIN and M. K. HORRIGAN

Dept of Metallurgy and Materials Science, University of Toronto, Toronto, M5S lA4, Canada

(Received 2 November 1983; in revised form 2 April 1984)

Abstract-Vapour pressure equations are presented for the metallic elements up to curium. For each elementa precise four-term equation and a more practically-based equation have been derived from a new evaluationofthermochemical data for the condensed and gaseous elements. It is intended that the equations should beused for ideal gases in the pressure range 10-15-10-3 atm, which covers the experimentally accessible range.The "precise" equations reproduce the data to better than 1% and the "practical" equations provide betterthan 5 % accuracy.

The vapour pressures of the metallic elements were previously

reviewed in the compilation of Hultgren et ai. [1] A critical

assessment of thermodynamic data has recently been completed

for the elements from literature data up to 1982 [2 J . These data

are for the condensed and gaseous states and hence the data for

any particular element can be combined to yield the vapour

pressure of the element as a function of temperature. These

equations are presented below for the 65 metallic elements up to

and including curium. The general equation which is employed

is a polynomial in temperature

contributions which influence the temperature dependence of

the thermal functions significantly.

In the examples which are given below for the metal

vanadium, the two-term equation is unacceptable, the three-

term equation is "practical", and the four-term equation is

"precise".

The vapour pressures listed in Table 1as input were calculated

using data from the recent evaluation for the chemical elements

logp(atm) = A + B T-1 + Clog T+ D T10-3

The fit of data to this polynomial form was made by least squares

analysis, and for each element, a two-, three- and four-term

equation was calculated in as wide a temperature interval as

seemed reasonable. Usually two temperature intervals only are

chosen, one for the solid and one for the liq uid. Based upon the

goodness-of-fit of the equations to the calculated vapour

pressures a "practical" equation with the minimum number of

terms has been selected which will reproduce the vapour

pressures to 5% or better. This criterion was se1ected as

representing a reasonable experimental error which would be

encountered in vapour pressure measurement, and which would

therefore serve most practical cases as well as reducing the

number of terms in the polynomial. The complete four-term

equation is also presented when the accuracy is improved, but it

is probably true that these equations are often over-precise when

the experimental uncertainties in the original thermochemical

data are taken into account.

The equations can, in principle, be used above the upper limit

of temperature, 2500 K, which is recommended here. It is felt

however that the thermal data for the condensed phases may

become significantly less accurate above 2300 K, and for gases,

new levels above the ground level may begin to make significant

309

Tempera-ture(K)

325424523622721820919

1018111712161315141415131612171118101909200821072206

Table 1. Vapour pressure of vanadium

log P (atm)

Calculated

Input Equation (1) Equation (2) Equation (3)

-75.109 -75.045 -75.121 -75.108-55.698 -55.684 - 55.692 -55.699-43.639 -43.653 -43.629 -43.641- 35.423 - 35.453 - 35.413 -35.424-29.466 -29.503 - 29.459 -29.466-24.950 -24.990 - 24.947 -24.949-21.409 -21.450 - 21.409 -21.408

-18.560 -18.598 - 18.563 -18.559-16.217 -16.252 - 16.223 -16.216-14.258 -14.287 -14.265 -14.257-12.596 -12.619 - 12.604 -12.596-11.168 -11.184 - 11.177 -11.168

-9.929 -9.937 -9.937 -9.930-8.844 -8.843 -8.851 -8.845-7.886 ......7.876 -7.892 -7.887-7.035 -7.014 -7.038 -7.036-6.274 -6.242 -6.273 -6.275-5.589 -5.546 -5.584 -5.590-4.971 -4.916 -4.961 -4.971-4.411 -4.342 -4.394 -4.408

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310 C. B. ALCOCK et al.: VAPOUR PRESSURES OF METALLIC ELEMENTS

[2]. The temperatures were chosen to illustrate the fact that the

polynomial form may be used to calculate the vapour pressure at

any selected temperature with any temperature interval. In this

case, the lowest temperature is arbitrarily set at 325 K and

calculations were made over 99 K intervals up to the melting

point of vanadium (Table 1).

The two-term vapour pressure equation was now calculated

using these data and the result is:

log p(atm) =7.87456 - 26948.9/7: (1 )

The three-term equation is

logp =9.74444 - 27132.3/T- 0.55006910g 7: (2)

The Equation (2) is satisfactory, according to our criteria to be

used as a "practical" equation, but a more precise fit is obtained

with the four-term equation

logp =

7.55339 - 27007.7/T+ 0.19699logT - 0.171188T/1000. (3)

It can be seen in Table 1that at most temperatures this equation

gives values which depart from the input values by more than

0.02. This is an acceptable difference between the input and the

calculated values, i.e. within the limits 5% of the vapourpressure.

The Equation (3) is sufficiently precise that no further terms

need be explored.

The origins of the data which have been used for this

calculation are indicated in the Tables 2 and 3.

Table 2. Recommended equations of the vapour pressures for the elements

logp(atm) =A + B 'T-1 + ClogT+ D T'1O-3

Elemert,

state A B

2 3

Li sol 5.667 -8310

Li liq 5.055 -8023

Na sol 5.298 -5603

Na liq 4.704 -5377

K sol 4.961 -4646

Kliq 4.402 -4453

Rb sol 4.857 -4215

Rb liq 4.312 -4040

Cs sol 4.711 -3999

Cs liq 4.165 -3830

Be sol 8.042 -17020

Be liq 5.786 -15731

Mg sol 8.489 -7813

Ca sol 10.127 -9517

Sr sol 9.226 -8572

Ba sol 12.405 -9690

Ba liq 4.007 -8163

Al sol 9.459 -17342

Alliq 5.911 -16211

Ga sol 6.657 -14208

Ga liq 6.754 -13984

In sol 5.991 -12548

In liq 5.374 -12276

Tl sol 5.971 -9447

Tlliq 5.259 -9037

Sn sol 6.036 -15710

Sn liq 5.262 -15332

Pb sol 5.643 -10143

Pb liq 4.911 -9701

Sc sol 6.650 -19721

Sc liq 5.795 -17681Y sol 9.735 -22306

Yliq 5.795 -20341

La sol 7.463 -22551

La liq 5.911 -21855

Ti sol 11.925 -24991

Ti liq 6.358 -22747

Zr sol 10.008 -31512

Zr liq 6.806 -30295

Hfsol 9.445 -32482

V sol 9.744 -27132

V liq 6.929 -25011

Nb sol 8.822 - 37 818

Ta sol 16.807 -41346

C

4

-0.4440

-0.8253

-1.4030

-1.1926

-2.2890

-0.7927

-0.3413

0.2885

-0.8705

-0.3142

-1.3376

-0.7890

-0.6735

-0.5501

-0.2575

-3.2152

D

5

-0.3663

0.7437

Temperature

range

6

298-m.p.

m.p.-l000

298-m.p.

m.p.-700

298-m.p.

m.p.-600

298-m.p.

m.p.-550

298-m.p.

m.p.-550

298-m.p.

m.p.-1800

298-m.p.

298-m.p.

298-m.p.

298-m.p.

m.p.-1200

298-m.p.

m.p.-1800

298-m.p.

m.p.-1600

298-m.p.

m.p.-1500

298-m.p.

m.p.-llOO

298-m.p.

m.p.-l 850

298-m.p.

m.p.-l 200

298-m.p.

m.p.-2000298-m.p.

m.p.-2300

298-m.p.

m.p.-2450

298-m.p.

m.p.-2400

298-m.p.

m.p.-2500

298-m.p.

298-m.p.

m.p.-2500

298-2500

298-2500

Refs

7

[2,4,8J

[2,4 J

[2-4,8J

[2-4J

[2, 4, 8 J

[2,4J

[2,4,8J

[2, 4J

[3,4,8J

[2-4 J[2,4,8 J

[2, 4 J

[3,4,8J

[2-4,8J

[3,4 J

[2-4J

[3,4 J

[2, 3, 8 J

[2,3 J

[2, 4, 6, 10J

[2,4J

[2,4 J

[2,4J

[2,4,6J

[2,4J

[2,4,8J

[2, 4J

[2, 4, 8 J

[2, 4J

[2, 4J

[2J

[2, 4J

[2, 6 J

[2,4,6J

[2, 4J

[2-4,11,12J

[2,3J

[2-5, 13J

[2, 3, 14 J

[2-4,15J

[2-4J

[3,4 J

[2,4J

[2,4 J

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C. B. ALCOCK et al.:VAPOUR PRESSURES OF METALLIC ELEMENTS 311

Table 2-cont.

Element, Temperaturestate A B C D range Refs

2 3 4 5 6 7

Cr sol 6.800 -20733 0.4391 -0.4094 298-2000 [2,4JMo sol 11.529 -34626 -1.1331 298-2500 [2-4 J

Wsol 2.945 -44094 1.3677 298-2350 [2,4JWsol -54.527 -57687 -12.2231 -2200-2500 [4 JMn sol 12.805 -15097 -1.7896 298-m.p. [2,6JRe sol 11.543 -40726 -1.1629 298-2500 [2, 16JFe sol 7.100 -21 723 0.4536 -0.5846 298-m.p. [2,3,6JFe liq 6.347 -19574 m.p.-2IOO [2JRu sol 9.755 -34154 -0.4723 298-m.p. [2 JOs sol 9.419 -41198 -0.3896 298-2500 [2,6JCO sol 10.976 -22576 -1.0280 298-m.p. [2,6,17JCO liq 6.488 -20578 m.p.-2150 [2JRh sol 10.168 -29010 -0.7068 298-m.p. [2 JRh liq 6.802 -26792 m.p.-2500 [2 JIr sol 10.506 -35099 -0.7500 298-2500 [2J

Ni sol 10.557 -22606 -0.8717 298-m.p. [2,3JNiliq 6.666 -20765 m.p.-2150 [2, 3JPd sol 9.502 -19813 -0.9258 298-m.p. [2, 18, 19JPd liq 5.426 -17899 m.p.-2100 [2 J

Pt sol 4.882 -29387 1.1039 -0.4527 298-m.p. [2,20JPt liq 6.386 -26856 m.p.-2500 [2JCU sol 9.123 -17748 -0.7317 298-m.p. [2,3,8JCU liq 5.849 -16415 m.p.-1850 [2,3,21 JAg sol 9.127 -14999 -0.7845 298-m.p. [2, 6, 8JAg liq 5.752 -13 827 m.p.-1600 [2JAu sol 9.152 -19343 -0.7479 298-m.p. [2 JAu liq 5.832 -18024 m.p.-2050 [2 JZn sol 6.102 -6776 298-m.p. [2, 3, 8JZn liq 5.378 -6286 m.p.-750 [2JCd sol 5.939 -5799 298-m.p. [2,6,8JCd liq 5.242 -5392 m.p.-650 [2 JHg liq 5.116 -3190 298-400 [2, 6, 8JCe sol 6.139 -21 752 298-m.p. [2,5,7JCe liq 5.611 -21200 m.p.-2450 [2,7,21 JPr sol 8.859 -18720 -0.9512 298-m.p. [2JPr liq 4.772 -17315 m.p.-2200

[2 JNd sol 8.996 -17264 -0.9519 298-m.p. [2,5,6JNd liq 4.912 -15824 m.p.-2000 [2,22JSm sol 9.988 -11 034 -1.3287 298-m.p. [2,23 JEu sol 9.240 -9459 -1.1661 298-m.p. [2,5,7JGd sol 8.344 -20861 -0.5775 298-m.p. [2,6JGd liq 5.557 -19389 m.p.-2250 [2 JTbsol 9.510 -20457 -0.9247 298-m.p. [2,5-7JTb liq 5.411 -18639 m.p.-2200 [2JDy sol 9.579 -15336 -1.1114 298-m.p. [2, 7, 24 JHo sol 9.785 -15899 -1.1753 298-m.p. [2, 5, 6 JEr sol 9.916 -16642 -1.2154 298-m.p. [2,6, 25, 26 JEr liq 4.688 -14380 m.p.-1900 [2 ]Tm sol 8.882 -12270 -0.9564 298-1400 [2, 6 JYb sol 9.111 -8111 -1.0849 298-900 [2,5,6JLu sol 8.793 -22423 -0.6200 298-m.p. [2,6JLu liq 5.648 -20302 m.p.-2350 [2 J

Th sol 8.668 -31483 -0.5288 298-m.p. [4,8,9JTh liq -18.453 -24569 6.6473 m.p.-2500 [2,9J

Pa sol 10.552 -34869 -1.0075 298-m.p. [7,9,27.28 ]

Pa liq 6.177 -32874 m.p.-2500 [9 JU sol 0.770 -27729 2.6982 -1. 5471 298-m.p. [2,4,5,8,9 JUliq 20.735 -28776 -4.0962 m.p.-2500 [4,9]

Np sol 19.643 -24886 -3.9991 298-m.p. [2,9JNpliq 10.076 -23378 -1.3250 m.p.-2500 [9 ]

Pu sol 26.160 -19162 -6.6675 298-600 [2,4,9,29JPu sol 18.858 -18460 -4.4720 500-m.p. [2,4,9,29JPu liq 3.666 -16658 m.p.-2450 [4,9JAm sol 11.311 -15059 -1.3449 298-m.p. [7,9,27JCm sol 8.369 -20364 -0.5770 298-m.p. [9JCmliq 5.223 -18292 m.p.-2200 [9 ]

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312 C. B. ALCOCK et al.: VAPOUR PRESSURES OF METALLIC ELEMENTS

Table 3. "Precisely fitted" equations of the vapour pressures for the elements

logp(atm) =A + B T-1 + Clog T+ D Tl0-3

Element, Temperature

state A B C D range

2 3 4 5 6

Li sol 7.790 -8423 -0.7074 298-m.p.

Li liq 8.409 -8320 -1.0255 m.p.-l000Na 1iq 8.400 -5634 -1.1748 m.p.-700

Kliq 8.233 -4693 -1.2403 m.p.-600

Rb 1iq 8.316 -4275 -1.3102 m.p.-550

Cs liq 8.232 -4062 -1.3359 m.p.-550

Be sol 4.933 -16869 0.6521 -0.3210 298-m.p.

Ca sol 4.751 -9278 0.5348 -0.6766 298-m.p.

Sr sol 4.809 -8385 0.4150 -0.5970 298-m.p.

Ba sol 8.403 -9524 -0.8280 -0.5551 298-m.p.

Al sol 7.124 -17246 0.0639 -0.3369 298-m.p.

Alliq 10.578 -16946 -1.3133 m.p.-1800

Ga 1iq 3.624 -13829 0.7579 -0.3141 m.p.-1600

In sol 8.532 -12680 -0.8495 298-m.p.

In liq 9.919 -12568 -1.5298 0.3377 m.p.-1500

TI sol 8.994 -9624 -0.9887 298-m.p.

Tl1iq 8.628 -9383 -1.0086 m.p.-l100

Sn sol 8.274 -15834 -0.7398 298-m.p.Sn liq 2.719 -15107 0.8036 -0.1033 m.p.-1850

Pb sol 8.336 -10303 -0.8782 298-m.p.

Pb liq 8.532 -10093 -1.0750 m.p.-1200

Y sol 6.421 -22138 0.2848 -0.3099 298-m.p.

Y liq 13.745 -22215 - 2.1235 m.p.-2300

La sol 7.514 -22553 -0.3323 0.0062 298-m.p.

La liq 6.524 -21977 -0.1667 m.p.-2450

Ti sol 7.704 -24772 0.1252 -0.3752 298-m.p.

Ti liq 16.370 -25229 -2.6574 m.p.-2400

Zr sol 7.779 -31376 -0.0354 -0.1658 298-m.p.

Zr liq 1.584 -28764 1.3555 m.p.-2500

Hfsol 8.936 -32449 -0.5041 -0.0331 298-m.p.

V sol 7.553 -27008 0.1970 -0.1712 298-m.p.

Nb sol 7.596 -37729 0.1441 -0.0706 298-2500

Mo sol 8.186 -34387 -0.0374 -0.1914 298-2500

Wsol -0.557 -43830 2.5145 -0.2074 298-2350

Wsol -110.184 -28414 35.4537 -3.6057 2200-2500Mn sol 6.504 -14794 0.4354 -0.6608 298-m.p.

Re sol 9.368 -40550 -0.4647 -0.1053 298-2500

Ru sol 5.944 -33895 0.7895 -0.2381 298-m.p.

Os sol 10.356 -41273 -0.6908 0.0459 298-2500

Co sol 6.436 -22348 0.5578 -0.4319 298-m.p.

Rh sol 8.721 -28923 -0.2168 -0.1076 298-m.p.

Ir sol 9.679 -35042 -0.4773 -0.0497 298-2500

Ni sol 9.397 -22547 -0.4663 -0.1114 298-m.p.

Pd sol 7.824 -19728 -0.3414 -0.1560 298-m.p.

Pt liq 21.257 -30921 -3.8960 m.p.-2500

Cu sol 7.810 -17687 -0.2638 -0.1486 298-m.p.

Cu liq 11.209 -17427 -1.4742 m.p.-1850

Ag sol 7.715 -14935 -0.2779 -0.1701 298-m.p.

Au sol 8.035 -19291 -0.3496 -0.1276 298-m.p.

Au liq 10.298 -18898 -1.2222 m.p.-2050

Zn sol 8.435 -6923 -0.7523 298-m.p.

Cd sol 8.405 -5944 -0.8052 298-m.p.Ce sol 7.340 -21834 -0.3851 0.0266 298-m.p.

Ce liq 6.023 -21278 -0.1127 m.p.-2450

Pr sol 5.167 -18556 0.3775 -0.4546 298-m.p.

Pr liq 11.917 -18693 -1.9565 m.p.-2200

Nd sol 4.542 -17061 0.6421 -0.5233 298-m.p.Nd liq 12.396 -17249 -2.0554 m.p.-2000Sm sol 3.881 -10753 0.8499 -0.6986 298-m.p.Eu sol 6.479 -9340 -0.1659 -0.3600 298-m.p.

Gd sol 5.158 -20706 0.5450 -0.3265 298-m.p.

Gdliq 10.606 -20505 -1.3598 m.p.-2250Tb sol 6.460 -20307 0.1472 -0.3066 298-m.p.Tb liq 11.803 -20052 -1.7216 m.p.-2200

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C. B. ALCOCK et aI.: VAPOUR PRESSURES OF METALLIC ELEMENTS 313

Table 3. cont.-

Element, Temperaturestate A B C D range

2 3 4 5 6

Dy sol 7.184 -15217 -0.2717 -0.2355 298-m.p.Ho sol 6.502 -15734 -0.0278 -0.3146 298-m.p.Er sol 7.669 -16529 -0.4312 -0.2116 298-m.p.Tm sol 7.062 -12185 -0.3098 -0.2019 298-1400Lu sol 7.347 -22347 -0.1187 -0.1288 298~m.p.Lu liq 18.851 -23540 -3.5093 m.p.-2350Th sol 11.954 -31682 -1.6430 0.2519 298-m.p.Th liq 59.574 -36981 -16.3130 1.9822 m.p.-2500Pa sol 4.601 -34501 1.0093 -0.4616 298-m.p.Pa liq 10.230 -34019 -1.0555 m.p.-2500Np sol 11.585 -24562 -1.0343 -1.1906 298-m.p.Np liq 16.266 -24018 -3.2613 0.2585 m.p.-2500Pu sol 18.958 -17609 10.5178 - 8.7185 298-600Pu sol 93.527 -22254 -30.9041 8.4953 500-m.p.Pu liq 12.992 -17587 -2.9370 0.4145 m.p.-2450Am sol 6.977 -14854 0.1914 -0.4699 298-m.p.Cm sol 3.571 -20128 1.1098 -0.4830 298-m.p.Cmliq 19.537 -21434 -3.8591 m.p.-2200

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