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. 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