JPEDAV (2015) 36:573–591
DOI: 10.1007/s11669-015-0411-5
1547-7037 ASM International
Thermodynamic Properties of Silver
J.W. Arblaster
(Submitted July 27, 2015; published online September 24, 2015)
The thermodynamic properties of silver have been evaluated to 2700 K. Selected values include
an enthalpy of sublimation of 284.8 ± 0.9 kJ/mol for the monatomic gas at 298.15 K, a dissociation enthalpy D0 of 157.7 ± 2.2 kJ/mol for the diatomic gas species at absolute zero, and a
derived equilibrium boiling point of 2433 K at one atmosphere pressure.
Keywords
gas, liquid, silver, solid, thermodynamic properties
1. Introduction
The freezing point is a primary fixed point on ITS-90 at
1234.93 K. (Preston-Thomas[1]). The equivalent thermodynamic temperature is currently considered to be about
0.05 K higher (Fischer et al.[2]). Wherever possible, values
have been corrected to the currently accepted atomic weight
of 107.8682 ± 0.0002 (Wieser et al.[3]) and to the ITS-90
temperature scale using correction factors of Douglas,[4]
Rusby,[5] Rusby et al.[6] and Weir and Goldberg.[7]
Previous reviews on silver have been given by Furukawa
et al.,[8] Hultgren et al.,[9] and CODATA (Cox et al.[10]).
Although the boiling point is below 2500 K thermodynamic
values were extended to 2700 K to take into account the
high temperature vapor pressure measurements of Geiger
et al.[11] (2049-2693 K).
2. Solid Phase
2.1 Range 0-4.2 K
Low temperature specific heat
given in terms
Pis generally
of the Debye equation: Cp ¼
a2nþ1 T 2nþ1 where below
n¼0
4.2 K only the first two terms are considered so that
Cp = cT + AT3 where c is the electronic coefficient and A is
usually represented in terms of a limited Debye temperature,
hD, where h3D = (12/5) p4 R/A = 1943.770/A where R is the
gas constant and A is given in units of J/mol K4. Most
values in this temperature range are given only in terms of
the 1958 Helium 4 temperature scale (Brickwedde et al.[12])
but a comparison with the ITS-90 Helium 4 temperature
scale (Preston-Thomas[1]) indicates that there is no straightElectronic supplementary material The online version of this
article (doi:10.1007/s11669-015-0411-5) contains supplementary
material, which is available to authorized users.
J.W. Arblaster, Wolverhampton, West Midlands WV5 8JU, England,
UK. Contact e-mail: jwarblaster@yahoo.co.uk.
forward relationship between the two scales and therefore it
would not be possible to preserve the simple two term
representation if such a conversion was used. Therefore,
values are as given on their original temperature scales. The
average values selected in Table 1 for c and hD are identical
with those selected by Phillips[13] but the value selected for
c is 0.003 mJ/mol K2 lower than that selected by Furukawa
et al.[8] reflecting the lower values obtained for later
measurements. In this region, only values obtained on metal
with a purity of at least 99.999% or described as being
‘‘spectroscopically pure’’ were considered in the evaluation
and included in Table 1. Other measurements on lower
purity materials were given by Furukawa et al.[8] Alers[14,15]
obtained a Debye temperature of 226.4 ± 0.6 K from elastic
constants measurements in satisfactory agreement with the
selected value of 226.0 ± 0.4 K obtained from thermodynamic measurements (Fig. 1 to 3).
2.2 Range 4.2-30 K
The specific heat measurements of Martin[30] (2.5-30 K)
were generally selected in this range but in order to
reconcile with the selected values of c and HD the latter
values were extended to 7 K whilst the value at 30 K
obtained by Martin[30] was rejected in favour of the later
value obtained from the specific heat measurements of
Martin[33] (20-300 K). The selected derived thermodynamic
data are included in Table 17 whilst the deviations of other
specific heat measurements in this region are given in
Table 14.
2.3 Range 30-298.15 K
The measurements of Martin[33] (20-320 K) were
selected because similar measurements on copper by
Martin,[34] in comparison with other high precision measurements, proved to be of very high quality. The measurements were given as an equation on the temperature scale
IPTS-1968 and were converted to ITS-90 using the
correction factors of Rusby.[5] However, rather than represent the revised values by the cumbersome sixteen coefficient equation as used by Martin[33] a series of lower order
equations were evaluated as given in Table 10. The selected
derived thermodynamic data are included in Table 17 whilst
the deviations of other specific heat measurements in this
region are given in Table 14, except for the mean specific
heat value of Richards and Jackson[35] (85-293 K). Above
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
573
Table 1
Electronic coefficient and Debye temperature values for silver
Authors
Ref
Temperature range, K
c, mJ/mol K2
HD, K
Notes
0.660
I 0.652
II 0.638
III 0.649
0.682
0.646
I 0.647
II 0.653
0.645
0.646
0.654
I 0.652
II 0.651
0.6450
0.644
0.645
0.6409
0.640
0.647 ± 0.005
225
I 226.5
II 223.4
III 229.0
226.2
225.5
I 226.2
II 226.5
228.9
226.6
225.8
I 225.4
II 225.7
226.0
226.2
226.3
227.3
226.6
226.0 ± 0.4
a
a, b
Keesom and Pearlman
Filby and Martin
[16]
[17]
1.2-2.4
0.4-1.5
Du Chatenier and De Nobel
Green and Culbert
Dixon et al.
[18,19]
[20]
[21]
2-30
2-4
1.2-4.2
Isaacs
Martin
Green and Valladares
Green
[22]
[23]
[24]
[25]
1.6-4.2
3-30
2-4
2-4
Ahlers
Montgomery et al.
Isaacs
Martin
Martin
Selected
[26]
[27]
[28]
[29]
[30]
1.4-26
1-4
1.6-4.2
0-30
2.5-30
I, II and III refers to the first, second and third sample respectively
a. Not included in the evaluation
b. Superseded by Martin[30]
c. Also Massalski and Isaacs[31] and Sargent et al.[32] Superseded by Isaacs[28]
Fig. 1
574
Low temperature specific heat of solid silver, taken from Table 17
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
a
a, c
a, b
a, b
Fig. 2 Specific heat of silver for 300 < T < 1500 K, calculated from expressions in Table 18
Fig. 3
High temperature thermodynamic properties of silver for 300 < T < 2800 K, taken from Table 18
30 K, the selected values of Furukawa et al.[8] essentially
represent a compromise between the discrepant specific heat
values of Bronson and Wilson[36] (193-393 K) and Meads
et al.[37] (14-298 K) and as such in this region trends from
0.7% high at 35 K to 0.1% low at 120 K then increasing to
an average of 0.4% high above 250 K. The subsequent
measurements by Martin[33] (20-300 K) showed agreement
to within 0.1% with the measurements of Bronson and
Wilson[36] in the overlap region whilst the measurements of
Meads et al.[37] trend to 1.1% high. Hultgren et al.[9] based
the low temperature measurements mainly on the specific
heat values of Eucken et al.[38] (11-205 K) and Meads
et al.[37] Above 20 K, these trend from 3.6% high initially to
1.4% low at 40 K then increase to an average of 0.6% high
above 200 K.
2.4 Range 298.15-1234.93 K
The enthalpy measurements of Cordfunke et al.[39] (558901 K) were selected since they not only extrapolated closely
to the selected specific heat value at 298.15 K but also because
the derived specific heat values averaged only 0.2% lower than
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
575
Table 2
Selected values for the solid at 298.15 K
Authors
Ref
Cp , J/mol K
H298:15
K H0 , J/mol
S298:15
K , J/mol K
Furukawa et al.
Hultgren et al.
This work
[8]
[9]
…
25.35
25.40
25.25
5745
5761
5734
42.56
42.68
42.48
the precision measurements of Bronson and Wilson[36]
(193-393 K). In order to extrapolate to the freezing point,
enthalpy measurements of Eastman et al.[40] (273-1173 K) at
973, 1073, and 1173 K were combined with the values of
Cordfunke et al.[39] to obtain Eq 1 which has an overall
accuracy as a standard deviation of ± 23 J/mol (0.20%) :
HT H298:15
K ðJ=molÞ ¼ 23:2782T þ 3:30338
103 T 2 7234:04
ðEq 1Þ
The values of Cordfunke et al.[39] were not corrected to
ITS-90 since the temperature scale used was not actually
given but the measurements of Eastman et al.[40] were
corrected from the contemporary atomic weight value of
107.88. Derived thermodynamic data are given in Table 18
whilst deviations of other high temperature specific heat
values are given in Table 14 and deviations of other
enthalpy values are given in Table 15. In the high temperature region, selected specific heat values of Hultgren
et al.[9] trend from initially 0.6% high to 1.2% low at 700 K
then increasing to 2.2% high at the freezing point whilst
selected values of CODATA (Cox et al.[10]) deviate from
0.4% high at 298.15 K to 0.9% low at 700 K increasing to
0.5% low at the freezing point.
2.5 A Comparison of Selected Values at 298.15 K
Both the low temperature values selected by Furukawa
et al.[8] and Hultgren et al.[9] are based on compromises
between discrepant specific heat values whilst CODATA
(Cox et al.[10]) accepted the values of Furukawa et al.[8] As
given in Table 2, the present evaluation is based on newer
high precision specific heat measurements which lead to
distinctly lower values of specific heat, enthalpy and entropy.
3. Liquid Phase
3.1 Enthalpy of Fusion
The procedure adopted by Stølen and Grønvold[41] is
accepted in that no values are rejected but are given a
percentage uncertainty (P) which is combined with the
enthalpy of fusion (DHM) to give a weight W1 = DHM P/100.
The contribution of each data point to the total is then given by
2
summations of these are given by
W2 = DH
P M/W1 and the P
XA = W2 and XB = 1/W12. The weighted average
value is then DHM = XA/XB and the standard deviation of
the fit is r = 1/XB. In Table 3, the first twelve values, which
were also given by Stølen and Grønvold,[41] have been
independently assessed but the assigned percentage uncertainties are those selected by the latter whilst for the remaining
576
values the percentage uncertainties were assigned in the
present evaluation. The enthalpies of fusion determined from
the measurements of Sommelet[42] and Corn[43] as given in
Table 3 were re-evaluations and compared to values originally given by the authors as 11,548 and 10,908 J/mol,
respectively. Based on more limited data sets, Hultgren
et al.[9] selected an enthalpy of fusion of 11.3 ± 0.4 kJ/mol
and CODATA (Cox et al.[10]) 11.0 kJ/mol, both values of
which can be considered to agree satisfactorily with the
presently selected value of 11.264 ± 0.109 kJ/mol.
3.2 Enthalpy and Specific Heat Values for the Liquid
The enthalpy of the solid at the freezing point combined
with the selected enthalpy of fusion leads to a fixed value of
37,815 ± 121 J/mol for the enthalpy HT H298:15K for the
liquid at the freezing point. A value for the specific heat of the
liquid is derived from a consideration of measurements of the
enthalpy with values being fitted to the equation
HT H298:15K = AT + B with the specific heat value derived
as the constant A. All values in Table 4 were recalculated to
derive the constants A and B with the measurements of
Sommelet,[42] Corn,[43] Feber et al.[60] and Sundareswaran
et al.[61] being corrected to ITS-90. The selected specific heat
value of 33.35 ± 0.46 J/mol K is a weighted average based on
N where N is the number of data points for each set of
measurements included in the evaluation. The combination of
the selected specific heat value with the selected value of the
enthalpy at the freezing point leads to Eq 2:
HT H298:15
K ðJ=molÞ ¼ 33:3500 T 3370:18
ðEq 2Þ
Derived thermodynamic properties to 2700 K are given
in Table 18. The liquid specific heat value of 32.1 J/mol K
determined by Vollmer and Kohlhaas[53] (1234-1400 K) is
notably lower than the selected value whilst an average
value obtained from the measurements of Wilde[62] (11101450 K) at 33.4 J/mol K is in satisfactory agreement.
Liquid specific heat values of 33.47 J/mol K selected by
Hultgren et al.[9] and 33.4 J/mol K selected by CODATA
(Cox et al.[10]) also agree satisfactorily with the selected
value. The deviations of the enthalpy values from the
selected equation are given in Table 16.
4. Gas Phase
4.1 Thermodynamic Properties of the Monatomic Gas
Selected values are based on the ten energy levels below
50,000 cm1 selected by Kraminda et al.[63] Thermody-
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
Table 3
Enthalpy of fusion values for silver
Authors
Ref
Methods
Person
Pionchon
Wüst et al.
Umino
Cavallaro
Wittig
Oelsen
Speros and Woodhouse
Dokken and Elliott
Vollmer and Kohlhaas
Callaghan
Callaghan
Kelley
Nathan and Leider
Sommelet
Corn
Nedumov
Orlik and Petrovin
Cagran et al.
Weighted average
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[54]
[55]
[56]
[42]
[43]
[57]
[58]
[59]
DC
DC
DC
DC
CC
DTA-EC
CC
DTA-Quant
DC
AS
DTA
DTA
PD
PD
DC
DC
DTA
Mod
RPH
DHM, J/mol
% Uncertainty
9513
11,161
11,748
11,242
10,903
10,926
10,924
11,418
12,096
11,400
11,510
11,693
11,632
11,632
11,331
11,142
10,732
12,186
10,958
11,264 ± 109
7.5
10
15
10
10
4
3
3
7.5
1.5
4
4
10
10
5
5
5
15
10
AS: adiabatic scanning; CC: cooling curves; DC: drop calorimetry; DTA: differential thermal analysis; DTA-Quant: quantitative differential thermal analysis;
DTA-EC: differential thermal analysis-electrical conductivity; Mod: modulation method; PD: phase diagrams; RPH: rapid pulse heating
Table 4
Enthalpy and specific heat values for liquid silver
Authors
Ref
Temperature range K
N
A
B
Cp , J/mol K
Notes
Wüst et al.
Umino
Sommelet
Corn
Feber et al.
Sunareswaran et al.
Cagran et al.
Weighted average
[46]
[47]
[42]
[43]
[60]
[61]
[59]
1248-1573
1273-1573
1241-1501
1235-1509
1244-1589
1281-1549
1234-2000
8
9
4
14
15
6
…
32.129
31.549
33.065
33.399
33.889
32.657
28.024
1666.44
+304.31
2898.06
3632.31
4574.18
2954.41
+278.38
32.13 ± 1.36
31.55 ± 0.80
33.07 ± 0.20
33.40 ± 0.50
33.89 ± 0.65
32.66 ± 0.74
28.02
33.35 ± 0.46
a
a
a
a. Not included in the evaluation
Table 5
Second law enthalpies of sublimation of the monatomic gas at 298.15 K
Authors
Ref
Method
Range, K
DH298:15
(II), kJ/mol
Jackson and Hudson
Ilschner and Humbert
Vintaikin et al.
Fedorov
De Maria and Malaspina
Avery et al.
Moore et al.
Golonka et al.
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
MS
MS
MS
KERT
KEMS
MS
MS
MS
1272-1346
1243-1473
1150-1220
924-1111
1121
1274-1552
1050
1102-1312
294 ± 3
284
291
276
274 ± 13
296 ± 8
277 ± 5
292 ± 2
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
577
Table 6
Enthalpies of sublimation of the monatomic gas at 298.15 K (values not included in the evaluation)
Ref
Method
Range, K
DH298:15
(II), kJ/mol
DH298:15
(III), kJ/mol
Von Watenberg
Hansen
Greenwood
Greenwood
Rosenhain and Ewen
Von Wartenberg
Ruff and Bergdahl
Jones et al.
Harteck
Farkas
Fischer
Fischer
Baur and Brunner
Schadel and Birchenall
Daane
Lyubimov and Granovskaya
Edwards and Downing
Nesmeyanov et al.
Knacke and Schmolke
Grieveson et al.
Kovtun et al.
Kučera et al.
Federov and Smirnov
Cochran and Foster
Kirshenbaun and Cahill
Fox and Esdale
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
[101]
[102]
[103,104]
[105]
BP
BP
BP
BP
Evap
Flow
BP
Evap
KE
Trans
BP
BP
BP
KERT
KE
KERT
Eff
KERT
Evap
Trans
Eff
KERT
KERT
KE
BP
Trans
Nachman et al.
Tomáš
Myles
Freeman
Haas and Schultze
Ansara and Bonnier
Matern
Haury
Andon et al.
Federov et al.
Schins et al.
Vatolin et al.
Fedichkin
Edwards
[106]
[107]
[108]
[109]
[110]
[111]
[112]
[113]
[114]
[115]
[116]
[117]
[118]
[119]
Novoselov et al.
Vaisburd et al.
Taberko and Vaisburd
Panday and Ganguly
Paule and Mandel: Lab 2
Paule and Mandel: Lab 8
Paule and Mandel: Lab 9
[120]
[121]
[122]
[123]
[69,70]
[69,70]
[69,70]
KE
Eff
TE
Miker
TE
KE
KE
KE
KE
KERT
BP
KE
Eff
KE
TE
TE
Eff
Eff
AA
CMS
KEMS
KE
2343
2313
2228
1923-2053
1143
1451-1708
1933-2213
1167-1234
1196-1344
1373
2423
1823-2298
1533-1819
1024-1234
1187-1334
1310-1840
1323-1513
830-1211
1048-1203
1523
1073-1233
1000-1176
1085-1229
1280-1383
2468
1113-1203
1113-1203
1251-1397
1024-1198
1128-1212
1150-1458
1205-1498
1292-1525
1157-1205
1372-1515
1175-1452
1113-1233
1965-2163
1300-1500
1078-1224
1160-1360
1160-1360
1282-1475
1253-1573
1248-1523
933-1053
854-1284
1144-1318
1254-1435
…
…
…
285
…
285 ± 15
282 ± 7
174 ± 24
283 ± 12
…
…
289 ± 10
245 ± 6
274
255 ± 13
305 ± 7
…
286 ± 8
274 ± 19
…
275
281 ± 1
305 ± 22
285 ± 8
…
269
274
278 ± 16
273 ± 11
269
310 ± 6
295 ± 1
288 ± 30
267
305 ± 7
272 ± 7
268
260 ± 9
286 ± 1
278
…
…
292
262 ± 6
265 ± 8
281
291 ± 1
269 ± 12
280 ± 23
275.1
272.0
263.0
263.4 ± 0.7
305.1 ± 0.8
283.8 ± 0.7
263.9 ± 0.5
307.0 ± 2.2
288.7 ± 0.4
284.5
283.7
286.5 ± 0.7
260.4 ± 0.5
281.8 ± 0.7
290.3 ± 0.7
303.5 ± 0.8
287.9
291.2 ± 0.8
278.7 ± 0.8
286.0
282.2 ± 0.5
294.9 ± 0.9
287.9 ± 0.9
288.9 ± 0.2
288.6
269.2 ± 0.2
271.4 ± 0.1
283.2 ± 0.5
281.0 ± 0.6
279.5 ± 0.4
287.0 ± 0.5
279.5 ± 0.1
283.8 ± 1.3
283.3 ± 0.3
284.2 ± 0.3
284.9 ± 0.5
274.0 ± 0.3
284.2 ± 0.2
289.8 ± 0.1
271.8 ± 0.4
283.5 ± 0.3
286.9 ± 0.3
286.3 ± 0.4
286.3 ± 0.8
285.5 ± 0.7
286.8 ± 0.4
281.5 ± 0.2
285.0 ± 0.4
277.4 ± 0.9
Authors
namic properties were calculated using the method of
Kolsky et al.[64] and the 2010 Fundamental Constants (Mohr
et al.[65,66]). Derived thermodynamic values are given in
Table 19.
578
Notes
a
a
a
a
a
a
a
b
c
b, d
a
b, e
b, e
b
b
b
a
b
f
f
b
b
g
h
4.2 Thermodynamic Properties of the Diatomic Gas
Thermodynamic properties were calculated by Rand[67]
from the six energy levels of X 1R+g , A 1R+u , B 1Pu, C 1Pu, D
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
Table 7
Enthalpies of sublimation of the monatomic gas at 298.15 K (values included in the evaluation)
Authors
Ref
Method
McCabe and Birchenall
McCabe et al.
[124]
[125]
KE
KE
Kornev and Vintaikin
Nesmeyanov et al.
Woolf et al.
Panish
[126]
[127,128]
[129]
[130]
KERT
KERT
Trans
KERT
Krupkowski and Golonka
Zavitsanos
O’Keefe
Boyer and Meadowcroft
Bohdansky and Schins
Tarby and Robinson
Vřeštál and Kučera
Marx et al.
Wachi et al.
Pomerantsov
[131]
[132]
[133]
[134]
[135,136]
[137]
[138]
[139]
[140]
[141]
Geiger et al.
Paule and Mandel:
Paule and Mandel:
Paule and Mandel:
Paule and Mandel:
Paule and Mandel:
Paule and Mandel:
Selected
[11]
[69,70]
[69,70]
[69,70]
[69,70]
[69,70]
[69,70]
L
KE
KE
KE
BP
Trans
KE
KE
KEMS
KE
KE
BP
KE
KE
KE
KE
TE
TE/KE
Lab
Lab
Lab
Lab
Lab
Lab
1
3
4
5
6
7
Range, K
DH298:15
(II), kJ/mol
DH298:15
(III), kJ/mol
1133-1273
(s) 1078-1228
(l) 1234-1336
994-1229
1040-1238
1522-1844
(s) 958-1228
(l) 1237-1503
1268-1462
1258-1575
1261-1310
1200-1500
1965-2150
1723-1823
1040-1202
1314-1583
1259-1546
(s) 1024-1221
(l) 1267-1354
2049-2693
1232-1585
1371-1531
1315-1585
1256-1519
1221-1434
1090-1209
285 ± 2
285 ± 6
279 ± 8
291 ± 3
276 ± 4
279 ± 1
291 ± 3
289 ± 12
287 ± 5
298 ± 5
293 ± 7
289
276 ± 4
281 ± 4
282 ± 6
294 ± 2
287 ± 3
296 ± 4
293 ± 3
285 ± 2
288 ± 2
296 ± 8
294 ± 2
285 ± 1
288 ± 1
282 ± 4
285.1 ± 0.1
285.1 ± 0.2
285.1 ± 0.2
286.1 ± 0.2
283.3 ± 0.2
284.2 ± 0.1
285.3 ± 0.3
283.7 ± 0.6
284.4 ± 0.2
285.3 ± 0.3
285.3 ± 0.1
284.9 ± 0.5
283.5 ± 0.2
284.7 ± 0.1
284.4 ± 0.3
286.2 ± 0.1
286.3 ± 0.2
284.7 ± 0.2
283.9 ± 0.1
284.9 ± 0.1
285.0 ± 0.1
284.3 ± 0.3
286.5 ± 0.1
285.6 ± 0.1
285.1 ± 0.1
283.1 ± 0.1
284.8 ± 0.9
Notes
b
a
a
Notes for Tables 5 to 7
DH298:15
(II) and DH298:15
(III) are the Second Law and Third Law enthalpies of sublimation at 298.15 K
a. Corrected for the presence of the diatomic gas
b. Given only as the Clausius-Clapeyron equation
c. Enthalpy of sublimation given only at 0 K
d. Also given by Amonenko et al.[142]
e. First measurement using nitrogen carrier gas and second measurement using argon carrier gas
f. Enthalpy of sublimation given only at 298.15 K
g. Data point at 1318 K rejected
h. Run 2 at 1284 K rejected
Methods for Tables 5 to 7
AA: atomic absorption; BP: boiling point; CMS: calibrated mass spectrometry; Eff: effusion; Evap: evaporation; Flow: dynamic flow; KE: Knudsen effusion;
KEMS: Knudsen effusion mass spectrometry; KERT: Knudsen effusion with radioactive tracer; L: Langmuir technique; Miker: Microbalance-inverted
Knudsen effusion-recoil; MS: mass spectrometry; TE: Torsion effusion; Trans: transport
R+u and E 1Pu for which full spectroscopic constants are
available (Beutel et al.[68]) except for the De values of C 1Pu
and D 1R+u which were estimated by Rand. Derived
thermodynamic data are given in Table 20.
1
4.3 Enthalpy of Sublimation of the Monatomic Gas
For values given in the form of the Clausius-Clapeyron
equation a ‘‘pseudo’’ Third Law value was calculated by
evaluating the enthalpy of sublimation at the temperature
extremes and then averaging. Because of a general lack of detail
as to what temperature scales were used, no attempt was made to
correct vapor pressure measurements to ITS-90 from what
would have been contemporary scales. Only high temperature
boiling determinations were corrected for the presence of the
diatomic gas. Values are summarised in Table 5 to 7.
The selected value of 284.8 ± 0.9 kJ/mol is an unweighted average of the measurements given in Table 7.
Seven sets of measurements included in Table 7 were also
selected by CODATA (Cox et al.[10]) who averaged to
284.9 ± 0.8 kJ mol1, although using the present thermodynamic values they would be averaged to
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
579
Table 8
Enthalpy of dissociation of the diatomic gas at 0 K
Authors
Drowart and Honig
Schissel
Ackerman et al.
Hilpert and Gingerich
Kingcade
Wilhite
Ran et al.
Weighted average
Ref
N
Range, K
D0 (II), kJ/mol
D0 (III), kJ/mol
[145,146]
[147]
[148]
[149]
[150]
[151]
[143]
…
14
7
13
31
8
123
1285
1275-1485
1336-1502
1402-1599
1219-1457
1197-1338
1121-1603
…
178 ± 11
186 ± 9
103 ± 19
177 ± 4
194 ± 10
…
163.3
156.3 ± 0.6
159.3 ± 0.6
155.7 ± 0.9
162.0 ± 0.2
156.2 ± 0.6
156.7 ± 0.6
157.7 ± 2.2
Notes
a
Notes for Table 8
D0 (II) and D0 (III) are the Second Law and Third Law enthalpies of dissociation at 0 K
a. Weighted average of eight runs
Methods for Table 8
Drowart and Honig[145,146] used mass spectrometry (MS). All other experiments used Knudsen effusion mass spectrometry (KEMS)
Table 9
Vapor pressure equations
Phase
Solid + Ag1
Solid + Ag2
Liquid + Ag1
Liquid + Ag2
Liquid + Ag1 + Ag2
Range, K
A
B
C
D
E
675-1234.93
600-1234.93
1234.93-2700
1234.93-2700
1234.93-2700
17.94147
28.70742
25.69820
44.08080
32.58274
0.299671
1.12684
1.51107
3.52974
2.48641
34378.2
49674.6
33913.5
48733.8
34503.1
3.973059104
7.382409104
+5.138389109
+4.764399105
+4.595059104
0
0
0
0
2.036099108
Table 10 Low temperature specific heat equations:
30-298.15 K
Range 30-50 K
Cp (J/mol K) = 1.48883 0.379934 T + 2.605509102
T2 3.794389104 T3 + 1.820479106 T4
Range 50-70 K
Cp (J/mol K) = 14.20335 + 0.845976 T 9.335539103
T2 + 6.725639105 T3 2.536139107 T4
Range 70-100 K
Cp (J/mol K) = 22.65606 + 1.19741 T 1.398369102
T2 + 8.272449105 T3 1.985689107 T4
Range 100-180 K
Cp (J/mol K) = 6.25383 + 0.544902 T 4.070999103
T2 + 1.465709105 T3 2.065579108 T4
Range 180-250 K
Cp (J/mol K) = 14.10307 + 0.630747 T 4.076419103
T2 + 1.213809105 T3 1.373639108 T4
Range 250-298.15 K
Cp (J/mol K) = 257.01381 + 4.18100 T 2.332519102
T2 + 5.789939105 T3 5.383349108 T4
284.5 ± 0.5 kJ mol1. Paule and Mandel[69,70] selected
284.55 ± 1.3 kJ mol1 as a weighted average of measurements obtained in nine laboratories.
580
4.4 Enthalpy of Dissociation of the Diatomic Gas
Free energy functions selected by Ran et al.[143] in the
range 600-1000 K trend from 0.4 to 0.6 J mol1 K1 lower
than the presently selected values. The reason for this is
unknown but the present evaluation is based on the
comprehensive review on spectroscopic constants by Beutel
et al.[68]. In agreement with Ran et al.[143] the gas Ag1/Ag2
pressure ratios were corrected for the variation of the
ionization cross section ratios rAg2/rAg1 with the experimental ionizing energies using the graphical representation
of Franzreb et al.[144] Values of the enthalpy of dissociation
summarised in Table 8 were weighted as N where N is the
number of data points. The mass spectrometric measurements of Drowart and Honig[145,146] were considered to be
preliminary and not included in the evaluation.
4.5 Vapor Pressure Equations
Values for the solid and Ag1 (g) were evaluated at 25 K
intervals from 675 to 1225 K and the freezing point; values
for the solid and Ag2 (g) at 50 K intervals from 600 to
1200 K and the freezing point. Values for the liquid and Ag1
(g) and for the liquid and Ag2 (g) were evaluated at 50 K
intervals form 1250 to 2700 K and the freezing point and
equilibrium values for liquid silver and Ag1 (g) + Ag2 (g)
by combining the two individual equations for the liquid.
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
Table 11
Values were fitted to the following equation with constants
given in Table 9:
High temperature representative equations
Solid: 298.15-1234.93 K
Cp (J/mol K) = 23.2782 + 6.606769103 T
HT H298:15
(J/mol) = 23.2782 T +
3.303389103 T2 7234.04
ST (J/mol K) = 23.2782 ln (T) +
6.606769103 T 92.1189
Liquid: 1234.93-2700 K
Cp (J/mol K) = 33.3500
(J/mol) = 33.3500 T 3370.18
HT H298:15
ST (J/mol K) = 33.3500 ln (T) 146.5377
Table 12
lnðp; barÞ ¼ A þ B lnðT Þ þ C=T þ DT þ ET 2
5. Summary of Representative Equations
Low temperature specific heat equations are given in
Table 10 and high temperature representative equations in
Table 11. Free energy equations are given in Table 12 and
transition values involved with the free energy equations in
Table 13.
Free energy equations above 298.15 K
Solid: 298.15-1234.93 K
GT H298:15
(J/mol) = 115.3971 T 3.303389103 T2
23.2782 T ln (T) 7234.04
Liquid: 1234.93-2700 K
GT H298:15
(J/mol) = 179.8877 T 33.3500 T
ln (T) 3370.18
6. Deviations from the Selected Values
Deviations of solid specific heat values are given in
Table 14. Deviations of enthalpy measurements for the solid
are in Table 15 and for the liquid in Table 16.
7. Thermodynamic Tables
Table 13 Transition values involved with the free energy equations
T, K
DHM
, J/mol
DSM
, J/mol K
1234.93
11264.000
9.1212
Transition
Fusion
Table 14
Low temperature thermodynamic properties of the solid
are given in Table 17 and of the high temperature thermodynamic properties of the condensed phases in Table 18.
Thermodynamic properties of the monatomic gas are given
Deviations of solid specific heat measurements
Authors
Ref
Temperature range, K
% deviations from the selected values
Nerst
Barschall
Brönsted
Griffiths and Griffiths
Eucken et al.
[152]
[153]
[154]
[155-157]
[38]
35-208
90
292
158-371
11-205
Keesom and Kok
[158]
1.4-20
Keesom and Kok
[159]
1.7-4.9
Moser
Bronson and Wilson
Meads et al.
[160]
[36]
[37]
325-925
193-393
14-298
Mustajoki
Lyashenko
Butler and Inn
Du Chatenier and De Nobel
[161]
[162]
[163]
[19]
328-758
373-1073
337-1090
1-30
Martin
[23]
3-30
Ahlers
Vollmer and Kohlhaas
[26]
[53]
5-26
300-1234
Scatters 4.1 low to 2.1 high
16.1 high
0.9 low
Trends from initially 1.6 high to 0.3 high at 302 K and to 0.6 high at 371 K
Shows scatter but can be considered as trending from initially 9.5 high to 2.1 low
at 20 K to 2.3 high at 43 K to 0.5 low at 205 K
Four runs but smooth values can be considered as trending 17.5 low at 1.4 K to
9.1 high at 10 K to 1.2 high at 20 K
Shows scatter but can be considered as trending from initially 7.0 low to 3.9 high
at 2.5 K to 9.8 low at 4.9 K
Average 2.4 low
Agrees to within 0.1-303 K then averages 0.2 high above 323 K
Nine runs but smooth values initially 4.8 high trending to 0.1 high at 90 K then
increasing to an average of 1.1 high above 250 K
General trend 0.1 high to 1.8 low
Trends 0.2 high to 0.4 low
General trend 8.4 low at 356 K to 7.4 high at 676 K to 3.8 low at 1090 K
Shows scatter but trends from initially 4.2 high to 0.5 high at 5 K to 2.6 high at 8 k
then to an average of 2.2 low above 21 K
Trends from initially 0.6 low to 0.6 high at 10 K to an average of 0.1 high above
21 K
Scatters 0.5 low to 0.4 high
Trends 1.8 low to 4.4 high
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
581
Table 15
Deviations of solid enthalpy measurements
Authors
Ref
Tilden
Magnus
Schimpff
Schübel
[164,165]
[166]
[167]
[168]
91-708
780-887
83-373
373-903
Weiss et al.
Wüst et al.
Eastman et al.
Umino
Magnus and Hodler
Roth and Bertram
[169]
[46]
[40]
[47]
[170]
[171]
618-859
373-1198
373-1173
373-1173
769-1179
373-1073
Jaeger et al.
[172,173]
374-1078
Jaeger et al.
Bronson et al.
Wittig and Böhm
Sommelet
Corn
Feber et al.
Dyunin et al.
Cagran et al.
[174]
[175]
[176]
[42]
[43]
[60]
[177]
[59]
374-1169
380-776
455-852
570-1226
1015-1230
1221-1230
573-798
1000-1234
Table 16
Temperature range, K
% deviations from the selected values
Above 373 K scatters 1.2 low to 1.8 low
Trends 0.7 high to 2.3 low
At 373 K 0.9 low
Trends 0.9 low to 2.7 low at 672 K to 1.2 low at
903 K
Trends 1.1 low to 0.5 low at 859 K
Scatters 5.5 low to 4.5 high
Trends from initially 0.3 high to converge to the selected values
Trends from initially 2.8 high to 5.6 high at 1173 K
Averages 4.3 low
General trend from initially 1.0 low to 1.6 low at
573 K to converge to selected value at 1073 K
Run 1 : Above 374 K trends 0.2 high to 0.7 low;
Run 2 : Above 694 K trends 0.7 low to 1.6 low
Averages 0.7 low
Trends from initially 0.4 high to 0.6 low at 776 K
Trends from initially 2.3 low to 0.1 low at 578 K then to 2.4 low at 852 K
Trends 2.9 low to 0.4 high
General trend 2.0 low to 0.2 low
Trends 1.5 low to 3.0 low
Averages 5.4 high
Trends from initially 1.9 high to 9.9 low
Deviations of liquid enthalpy measurements
Authors
Ref
Temperature range, K
% deviations from the selected values
Wüst et al.
Umino
Sommelet
Corn
Feber et al.
Sundareswaran et al.
Cagran et al.
[46]
[47]
[42]
[43]
[60]
[61]
[59]
1248-1573
1273-1573
1241-1501
1235-1509
1244-1589
1281-1549
1234-2000
Scatter 1.8 low to 1.2 high
Trends 3.5 high to 1.5 high
Averages 0.2 high
General trend 0.5 high to 1.3 low at 1247 K to average 0.4 low above 1405 K
Scatter 2.0 low to 0.7 high
Scatters but averages 1.3 low
Trends 7.8 low to 11.1 low
Table 17
Low temperature thermodynamic properties
Temperature, K
5
10
15
20
25
30
35
40
45
50
60
70
80
90
100
110
582
Cp , J/mol K
HT H0 , J/mol
ST , J/mol K
GT H0 , J/mol
GT H0 T, J/mol K
0.0243
0.182
0.670
1.651
3.076
4.770
6.572
8.356
10.042
11.578
14.188
16.251
17.863
19.122
20.117
20.911
0.0344
0.461
2.404
7.998
19.66
39.21
67.55
104.9
150.9
205.1
334.4
487.0
657.9
843.1
1039
1245
0.0103
0.0634
0.215
0.530
1.044
1.752
2.622
3.617
4.700
5.839
8.189
10.538
12.818
14.997
17.066
19.022
0.0169
0.173
0.813
2.594
6.444
13.36
24.24
39.79
60.55
86.88
157.0
250.7
367.5
506.7
667.1
847.6
0.00337
0.0173
0.0542
0.130
0.258
0.445
0.692
0.995
1.346
1.738
2.616
3.581
4.594
5.630
6.671
7.706
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
Table 17
continued
Temperature, K
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
273.15
280
290
298.15
Table 18
HT H0 , J/mol
ST , J/mol K
GT H0 , J/mol
GT H0 T , J/mol K
21.556
22.086
22.525
22.895
23.211
23.486
23.725
23.934
24.116
24.280
24.430
24.570
24.697
24.807
24.902
24.996
25.026
25.094
25.188
25.248
1457
1675
1899
2126
2356
2590
2826
3064
3305
3547
3790
4035
4281
4529
4778
5027
5106
5277
5529
5734
20.870
22.617
24.270
25.837
27.325
28.741
30.090
31.378
32.611
33.791
34.924
36.013
37.062
38.072
39.047
39.989
40.279
40.899
41.782
42.481
1047
1265
1499
1750
2016
2296
2590
2898
3218
3550
3893
4248
4613
4989
5375
5770
5896
6174
6588
6931
8.726
9.728
10.708
11.665
12.598
13.506
14.390
15.251
16.088
16.903
17.697
18.470
19.223
19.957
20.672
21.370
21.587
22.052
22.717
23.247
High temperature thermodynamic properties
Temperature, K
298.15
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1200
1234.93
1234.93
1300
1400
1500
1600
1700
1800
1900
Cp , J/mol K
Cp , J/mol K
HT H298:15
, J/mol
ST , J/mol K
GT H298:15
T, J/mol K
25.248
25.260
25.591
25.921
26.251
26.582
26.912
27.242
27.573
27.903
28.233
28.564
28.894
29.224
29.555
29.885
30.215
30.546
30.876
31.206
31.437
33.350
33.350
33.350
33.350
33.350
33.350
33.350
33.350
0
46.7
1318
2606
3910
5231
6568
7922
9292
10,679
12,083
13,503
14,939
16,392
17,862
19,348
20,850
22,369
23,905
25,457
26,551
37,815
39,985
43,320
46,655
49,990
53,325
56,660
59,995
42.481
42.637
46.556
49.994
53.066
55.849
58.398
60.754
62.948
65.003
66.940
68.772
70.514
72.175
73.764
75.288
76.754
78.167
79.532
80.853
81.752
90.873
92.586
95.057
97.358
99.511
101.532
103.439
105.242
42.481
42.481
42.790
43.480
44.377
45.388
46.456
47.551
48.652
49.747
50.829
51.894
52.938
53.961
54.962
55.940
56.897
57.832
58.746
59.640
60.252
60.252
61.828
64.115
66.255
68.267
70.165
71.961
73.666
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
583
Table 18
continued
Temperature, K
2000
2100
2200
2300
2400
2500
2600
2700
Table 19
Cp , J/mol K
HT H298:15
, J/mol
ST , J/mol K
T , J/mol K
GT H298:15
33.350
33.350
33.350
33.350
33.350
33.350
33.350
33.350
63,330
66,665
70,000
73,335
76,670
80,005
83,340
86,675
106.952
108.580
110.131
111.613
113.033
114.394
115.702
116.961
75.287
76.834
78.313
79.729
81.087
82.392
83.648
84.859
Thermodynamic properties of the monatomic gas
Temperature, K
298.15
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1200
1234.93
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
Cp , J/mol K
HT H298:15
, J/mol
ST , J/mol K
GT H298:15
T, J/mol K
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.786
20.787
20.787
20.788
0
38.5
1078
2117
3156
4196
5235
6274
7314
8353
9392
10,432
11,471
12,510
13,549
14,589
15,628
16,667
17,707
18,746
19,472
20,825
22,903
24,982
27,060
29,139
31,218
33,296
35,375
37,454
39,532
41,611
43,689
45,768
47,847
49,925
172.997
173.126
176.330
179.106
181.554
183.744
185.725
187.534
189.198
190.738
192.172
193.514
194.774
195.962
197.086
198.152
199.166
200.133
201.057
201.942
202.538
203.605
205.146
206.580
207.921
209.182
210.370
211.494
212.560
213.574
214.541
215.465
216.350
217.198
218.013
218.798
172.997
172.998
173.251
173.813
174.540
175.353
176.207
177.077
177.946
178.805
179.649
180.474
181.279
182.062
182.823
183.563
184.282
184.981
185.660
186.320
186.770
187.586
188.786
189.925
191.009
192.041
193.027
193.969
194.872
195.739
196.572
197.373
198.146
198.891
199.611
200.307
H298:15
H0 6197.4 J/mol
584
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
Table 20
Thermodynamic properties of the diatomic gas
Temperature, K
Cp , J/mol K
HT H298:15
, J/mol
ST , J/mol K
=T, J/mol K
GT H298:15
37.073
37.081
37.264
37.398
37.503
37.590
37.664
37.729
37.789
37.844
37.896
37.945
37.993
38.039
38.084
38.128
38.171
38.214
38.256
38.297
38.326
38.379
38.460
38.540
38.620
38.698
38.777
38.855
38.933
39.011
39.089
39.167
39.246
39.324
39.404
39.484
0
68
1927
3794
5666
7544
9425
11,310
13,198
15,089
16,982
18,878
20,777
22,678
24,581
26,486
28,393
30,303
32,215
34,129
35,467
37,963
41,804
45,654
49,512
53,378
57,252
61,134
65,023
68,920
72,825
76,738
80,659
84,587
88,524
92,468
257.348
257.578
263.308
268.294
272.705
276.661
280.247
283.527
286.549
289.352
291.965
294.412
296.714
298.887
300.945
302.899
304.760
306.537
308.237
309.866
310.965
312.934
315.782
318.438
320.928
323.271
325.486
327.584
329.579
331.481
333.297
335.037
336.705
338.309
339.853
341.341
257.348
257.349
257.801
258.808
260.112
261.572
263.109
264.676
266.244
267.796
269.321
270.813
272.270
273.689
275.070
276.413
277.719
278.988
280.223
281.425
282.245
283.732
285.921
288.001
289.982
291.872
293.679
295.408
297.067
298.661
300.195
301.672
303.097
304.474
305.805
307.094
298.15
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1200
1234.93
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
H298:15
H0 10; 215 J=mol
Table 21
Vapor pressure
Monatomic gas
T, K
p bar
298.15
300
400
500
600
700
8.3791044
1.7091043
4.1491031
1.0991023
9.3891019
3.0991015
Diatomic gas
DGT , J/mol
DHT , J/mol
p bar
DGT , J/mol
DHT , J/mol
245,886
245,645
232,667
219,818
207,084
194,459
284,800
284,792
284,311
283,765
283,152
282,474
1.6891063
4.6591063
2.9791045
1.3291034
1.5491027
1.6591022
358,323
358,000
340,981
624,322
307,975
291,909
409,720
409,695
408,302
406,802
405,186
403,450
Journal of Phase Equilibria and Diffusion Vol. 36 No. 6 2015
585
Table 21
continued
Monatomic gas
T, K
p bar
800
900
1000
1100
1200
1234.93
1234.93
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
1.3291012
1.4591010
6.179109
1.319107
1.669106
3.659106
3.659106
1.349105
7.709105
3.499104
1.309103
4.129103
1.159102
2.859102
6.439102
0.134
0.260
0.475
0.824
1.364
Diatomic gas
DGT , J/mol
DHT , J/mol
p bar
DGT , J/mol
DHT , J/mol
181,936
169,510
157,177
144,936
132.783
128,559
128,559
121,314
110,259
99,294
88,413
77,610
66,852
56,223
45,630
35,101
24,631
14,218
3860
6446
281,729
280,918
280,041
279,098
278,089
277,721
266,457
265,640
264,383
263,127
261,871
260,614
259,358
258,102
256,845
255,589
254,333
253,076
251,820
250,563
9.3991019
7.5891016
1.9491013
1.1991011
4.3291010
1.319109
1.319109
7.939109
8.929108
7.159107
4.369106
2.129105
8.579105
2.969104
8.949104
2.419103
5.919103
1.339102
2.789102
5.459102
276,100
260,530
243,388
230,063
215,145
209,982
209,982
201,622
188,951
176,484
164,203
152,098
140,157
128,374
116,736
105,236
93,868
82,627
71,505
60,497
401.593
399,614
397,511
395,285
392,936
392,086
369,558
367,713
364,884
362,064
359,252
356,448
353,652
350,864
348,083
345,310
342,545
339,788
337,039
334,297
DH0 (Ag1) 284.337 kJ/mol; DH0 (Ag2) 410.974 kJ/mol
Table 22
Acknowledgment
Equilibrium vapor pressure
Total p bar
Ag1 p bar
Ag2 p bar
T, K
1015
1014
1013
1012
1011
1010
109
108
107
106
105
104
103
102
101
1
NBP
1.0091015
1.0091014
1.0091013
1.0091012
1.0091011
1.0091010
1.009109
1.009108
1.009107
1.009106
9.999106
9.999105
9.979104
9.939103
9.849102
0.9675
0.9802
1.2391024
3.3091023
8.8591022
2.3791020
6.3191019
4.4691016
1.1891014
3.1091013
8.1091012
2.1191010
6.329109
1.469107
3.309106
7.339105
1.589103
3.259102
3.309102
684
717
754
795
840
891
949
1015
1090
1179
1285
1416
1579
1785
2057
2430.55
2433.10
NBP, normal boiling point at one atmosphere pressure (1.01325 bar)
in Table 19 and of the diatomic gas in Table 20. The vapor
pressure summary is given in Table 21 and equilibrium
vapor pressure data in Table 22.
586
The author is indebted to Malcolm Rand for calculating
the thermodynamic properties of the diatomic gas.
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