Chapter 4. Standard entropies of hydration of ions

摘要

4 Standard Entropies of Hydration of Ions By Y. MARCUS and A. LOEWENSCHUSS Department of Inorganic and Analytical Chemistry The Hebrew University of Jerusalem 9I904Jerusalem Israel 1 Introduction Since ions are hydrated in aqueous solutions the standard thermodynamic functions of the hydration process are of interest. Of these the standard entropy of hydration is expected to shed some light on the state of the ion and the surrounding aqueous medium. In particular the notion of water-structure-breaking and -making is amenable to quantitative expression in terms of a structural entropy that can be derived from the experimental standard molar entropy of hydration. The process of hydration of an ion X' where z is the algebraic charge number of the ion consists of its transfer from the ideal gas phase to the aqueous phase at infinite dilution X'(g) (1) -X'(aq) A thorough discussion of the general process of solvation of which hydration is a particular case was recently published by Ben-Naim and Marcus,' with a sequel on the solvation of dissociating electrolytes by Ben-Naim.2 Provided that the number density concentration scale or an equivalent one (e.g.the molar i.e. mol dm-') is employed the standard molar entropy change of the process conventionally determined experimentally and converted appro- priately to an absolute value equals Avogadro's number NA times the entropy change per particle.'-2 The conventional standard states are the ideal gas at 0.1 MPa pressure (formerly at 0.101 325 MPa = 1 atm pressure) for the gas phase (g) and the ideal aqueous solution under 0.1 MPa pressure and at 1 mol dmP3 concentration of the ion for the aqueous phase.Most discussions are limited to the entropies of hydration at 298.15 K although the entropies of hydration at other temperatures in particular elevated temperatures are expected to provide interesting information too. Recently a large amount of information was published that is pertinent to ' A. Ben-Naim and Y. Marcus J. Chem. Phys. 1984 81 2016. * A. Ben-Naim J. Phys. Chem. 1985 81 in the press. 81 Y. Marcus and A. Loewenschuss the present topic in a compilation by Wagman et aL3 and in a review by Loewenschuss and Mar~us.~ The former contains conventional standard partial molar entropies of aqueous ions at 298.1 5 K the latter the standard molar entropies of polyatomic gaseous ions again at 298.15K.These sources supplemented by the readily calculated standard molar entropies of monoatomic gaseous ions provide the bulk of the information presented and discussed in the present report. These data are further supplemented by data from other sources or by values estimated in the present work. Sources have been scanned through the years 1979 to 1983 inclusive and further back where necessary. A previous survey of the entropy of hydration of ions that may be con- sulted is that of Friedman and Krishnan,' who discussed this subject within the framework of a discussion of the thermodynamics of ion hydration. 2 Conventional Entropies of Hydration at 298.15K The Available Data.-The conventional standard molar entropy of hydration is reported in Table 1 for nearly 200 ions. This quantity is given in view of equation (l) by -0 AhydrConv = amp;o,,(as) -SPd (2) where the subscript i stands for an ion X' and the quantities on the right hand side are defined below. The conventional standard partial molar entropy of the aqueous ion sEon,(aq) is obtained from the actual standard molar entropy change A(3)So for the reaction (3) which is accessible experimentally. The hypothetically ideal state of 1 mol (kgwater)-' is generally used for the aqueous ions the pure substance (unionized) in its standard state (ss) for X(ss) and the ideal gas state for the hydrogen all at the standard pressure of 0.1 MPa.The convention that SPconv(aq)= 0 for H+(aq) is then employed to convert the A(3,S0value to a value of $amp;(aq) for the ion XZ(aq). The conversion from the 1 mol (kg water)-' (molal) concentration scale to the 1 moldm-3 (molar) one is done by noting that 1 kg water occupies 1.002964 dm3 at 298.15 K,6hence the amount -R In (1.002964 dm3/l dm3) = 0.024 J K-' mol-' must be added to the con- ventional values on the molal scale (tabulated e.g. in ref. 3). This correction is entirely negligible in view of the accuracy and precision claimed for the reported sEonv (as) data. The uncertainties of these values reported in Table 1 follow the code of being 8 to 80 times the unit of the last digit rep~rted.~ D. D. Wagman W. H.Evans V. B. Parker R. H. Schumm I. Halow S. M. Balley K. L. Churney and R. L. Nuttal 'The NBS Tables of Chemical Thermodynamic Properties' American Chemical Society and American Institute of Physics Washington DC 1982. A. Loewenschuss and Y. Marcus Chem. Rev. 1984 84 89. H. L. Friedman and C. V. Krishnan in 'Water A Comprehensive Treatise' ed. F. Franks Plenum Press New York NY Vol 3 1973. K. S. Kell J. Chem. Eng. Dara 1975 20 97. Standard Entropies of Hydration of Ions Table 1 Conventional standard molar entropies of hydration of ions at 298.15 K arranged according to the order in the NBS Tables (in J K-' mol-') No. 0 1 2 3 4 Ion e-O2-O2-0;-H+ * * * * * s?(g) 20.98 143.32 203.8 199.6 108.84 siv (aq) 35.2 -86. 100. -100. 0.00 ~hydr$on 14.2 -229.-104. -300. -108.84 5 6 OH-H30+ * 172.3 192.8 0.00 -10.7 -183.0 -192.8 7 HOT 228.6 23.8 -204.8 8 F- 145.59 -13.8 -159.4 9 10 HF;c1- 21 1.3 154.40 92.5 56.5 -118.8 -96.9 11 c10- 215.7 42. -174. 12 ClO 257.0 101.3 -155.7 13 ClO; 264.3 162.3 -102.0 14 clod 263.0 184.0 -79.0 15 Br- 163.57 82.4 -81.2 16 Bry 326.6 215.5 -111.0 17 BrO- 227.2 42. -185. 18 BrO; 278.7 161.7 -117.0 19 BrO; 282.1 199.6 -82.5 20 I- 169.36 111.3 -58.1 21 22 1;IOj 334.7 288.2 239.3 118.4 -95.4 -169.8 23 23a 10; At - * 297.0 167.47 222. 126. -75. -41. 24 S2- 152.14 -14.6 -166.7 25 26 s;-s ~ 223.1 285.4 28.5 66.1 -194.6 -219.3 27 so:~ 264.3 -29. -293. 28 29 so;-s20:- 263.6 291.1 18.8 67. -244.8 -224. 30 31 32 33 S,O,z-s,o;~-s20;-s,o;- * * 319. 337.3 341.356. 92. 125. 244.3 257.3 -227. -212. -97. -99. 34 HS - 186.2 66. -120. 35 HSO; 266.8 139.7 -127.1 36 37 HSO Se2- * 283.0 163.42 131.8 0. -151.2 -163. 38 SeO:- 284.0 13. -271. 39 SeO:- 28 1.2 54.0 -227.2 40 HSe- 203.8 80. -124. 41 HSeO; 283.0 135.1 -147.9 42 HSeO; 295.8 149.4 -146.4 43 44 TeO -TeOi- * 294.5 295.7 13.4 46. -281.1 -250. 45 46 NO+N; * 212.2 198.4 107.9 -103. -104.3 -301. 47 NO 214.1 -93. -307. 48 NO; 236.3 123.0 -113.3 49 NO; 245.2 146.6 -98.6 84 Y. Marcus and A. Loewenschuss No. Ion SP(g) con"(aq) Ahydr sp0fl" 50 N20i-* 256.9 28. -229. 51 NH 186.3 96.9 -89.4 52 N2H 230.5 151. -80. 53 N2Hi+ * 225.2 79. -146. 54 NH,OH+ * 235.4 155. -80. 55 Po -266.4 -222. -488.0 56 P,O' -342.8 -117.-460. 57 HPOi-283.0 -33.5 -3 16.5 58 H PO; 280.7 92.5 -188.2 59 AsO; * 268.2 40.6 -227.6 60 AsOj-282.9 -168.9 -451.8 61 HASO:-302.9 -1.7 -304.6 61a SbO+ * 23 1.6 22. -210. 62 Sb0:-* 298.5 -155. -454. 63 Bi3+ * 175.90 -151.8 -327.7 64 co:-246.1 -43.5 -289.6 65 c,o -295.1 45.6 -249.5 66 HCO; 238.2 92. -146. 67 HCO 257.9 98.4 -159.5 68 CH3CO; 278.7 86.6 -192.1 69 CN-196.7 94.1 -102.6 70 CNO-218.9 106.7 -112.2 71 SCN-232.5 144.3 -88.3 72 CH3NH 327.7 142.7 -90.0 73 (CH3),N+ * 331.9 210. -122. 74 (C,H,)4N+ * 483. 283. -200. 75 (C3H7)4N+* 641* 336. -305. 76 SiFi-309.9 122.2 -187.7 77 Sn2 168.52 -17. -186. + 78 Sn4 168.52 -117. -286. + 79 SnFz-* 354.0 220. -134. 80 Pb2+ 175.49 10.5 -165.0 81 BO; 215.8 -37.2 -253.0 82 BH; 187.7 110.5 -77.2 83 BF; 267.9 180.-88. 84 B(OH); 270.5 101.2 -169.3 85 ~13 149.98 -321.7 -471.7 + 86 A10; 229.5 -8.8 -238.3 87 Al(0H); 293. 24.6 -268. 88 Ga3+ 161.86 -331. -493. 89 1~3+ 168.11 -151. -319.1 90 T1+ 175.32 125.5 -49.8 91 ~13+ 175.32 -192. -367. 92 Zn2+ 161.06 -112.1 -273.2 93 Cd2+ 167.84 -73.2 -241.0 94 Hg2+ 175.09 -32.2 -207.3 95 Hg:+ 273.0 65.5 -207.5 96 cu+ 161.05 40.6 -120.5 97 cu2 175.95 -99.6 -275.5 + 98 Ag+ 167.44 72.68 -94.76 99 AgC1; 290.7 231.4 -59.3 100 Ag(NH3) * 241.1 245.2 (+4.1) 101 Ag(CN); 284.5 172. -113. 102 AuCl; 363.8 266.9 -96.9 103 Au(CN); 284.5 172. -113. I04 Ni2-178.05 -128.9 -306.9 Standard Entropies of Hydration of Ions -0 No. Ion so(g) sicon" (ad Ahydr SPonv 105 Co2+ 179.52 -113.-293. * 106 co3 179.30 -305. -484. + 107 Co(NH3);+ 435.2 146.0 -289.2 108 Co(CN)i-464.8 232.6 -232.2 109 Fe2+ 180.32 -137.7 -318.0 110 Fe3+ 173.99 -315.9 -489.9 111 Fe(CN)i-482.5 270.3 -212.2 112 Fe(CN):-469.8 95.0 -374.8 113 Pd2+ 185.42 -184. (-369.) 114 PdC1;-412.2 272. -140. + 115 Pd( NH ) 4 10.0 307. -103. 116 RhC1;-410.4 209. -201. 117 Pt2+ 194.6 -79. -274. 118 ptc1;-425.7 218.7 -207.0 119 Pt(NH3):+ 415.8 44.8 (-37 1.O) 120 IrCIi-421.9 222. -200. 121 IrCli-4 16.7 180. -237. 122 Mn2+ 173.78 -73.6 -247.4 123 MnO; 277.8 191.2 -86.6 124 MnOi-291.1 58.6 -232.5 125 TcO; 288.5 197.5 -91.0 * 126 Re-183.30 230. (+47.) 127 ReO 294.1 201.3 -92.7 128 ReClg-426.8 61. -366.* 129 Cr2+ 181.74 -82. -264. * + 130 Cr3 178.96 -269. -448. 131 CrO:-28 1.4 50.2 -231.2 132 Cr20:-379.7 261.9 -117.8 133 MOO:-291.1 27.2 -263.9 134 wo 196.6 40.6 -256.0 * 135 V2 182.09 -74. -256. * + 136 v3+ 177.84 -307. -485. + 137 vo2 225.9 -133.9 -359.8 138 vo 259.0 -42.3 -301.3 139 vo 266.9 50. -217. 140 vo -* 284.8 -172. -457. 141 HVOi-296.1 17. -279. * + 142 Zr4 165.23 -509.3 -674.5 * 143 HP+ 173.63 -465.7 -630.3 144 sc3+ 156.37 -255. -411. 145 Y3-164.90 -251. -416. 146 Lu3+ 173.38 -264. -437. * 147 Yb2+ 173.24 -47. -220. 148 Yb3+ 190.53 -238. -429. 149 Tm3+ 194.26 -243. -437. 150 Er3+ 195.87 -244.3 -440.2 151 HO~+ 196.19 -226.8 -423.0 152 Dy3+ 195.50 -231.0 -426.5 153 Tb3' 193.50 -226.-420. 154 Gd3+ 189.33 -205.9 -395.2 155 Eu2+ 188.90 -8. -197. + 156 Eu3 180.48 -222. -402. * + 157 Sm2 183.12 -26. -209. + 158 Sm3 186.75 -21 1.7 -398.5 * + 159 Pm3 189.46 -210. -399. 86 Y. Marcus and A. Loewenschuss No. Ion SP (g) (ad $l" 160 Nd3+ 190.10 -206.7 -396.8 + 161 or3 188.93 -209. -398. + 162 Ce3 185.49 -205. -390. 163 Ce4+ 170.60 -301. -472. + 164 La3 170.49 -217.6 -388.1 * 165 Bk3+ 199.O -187.1 -386.1 * 166 Bk4+ 195.0 -395. -590. * + 167 Cm3 194.8 -188.4 -383.2 * 168 Am3+ 177.4 -203.9 -381.3 * 169 AmO 278. -20.9 -299. * 170 AmOi-274. -88. -362. 171 PU'+ 192.18 -186.1 -378.3 * 172 Pu4+ 190.66 -389. -580. * 173 Puo; 275. -20.9 -296. + I74 puo; 270. -87.9 -358.175 Np3' * 195.59 -181.0 -376.6 * 176 Np4+ 188.71 -389. -578. 177 NPO 272. -20.9 -293. 178 NpO:+ 266. -92.0 -358. 179 u3 * * 196.48 -176.5 -373.0 + 180 u4+ 186.36 -414. -600. 181 uo 269. -29. -298. 182 uo:+ 260.4 -98.3 -358.7 183 Th4+ 176.91 -422.6 -599.5 * 184 Ac3+ 176.63 -184.4 -361.0 185 Be2+ 136.26 -129.7 -266.0 186 Mg2+ 148.68 -138.1 -286.8 187 Ca2 154.93 -53.1 -208.0 + 188 Sr2+ 164.72 -32.6 -197.3 189 Ba2+ 170.35 9.6 -160.75 190 Ra2+ 176.58 54. -123. 191 Li + 132.99 13.4 -119.6 192 Na+ 147.98 59.0 -89.0 I93 K+ 154.63 102.5 -52.1 I94 Rb+ 164.41 121.50 -42.91 195 cs+ 169.86 133.05 -36.81 The standard molar entropies of gaseous monoatomic ions are the sums of the translational entropy and the contributions of the 'magnetic' and 'elec- tronic' entropies from unpaired electron spins and the population of higher electronic levels at the given temperature.The former is obtained from the Sackur-Tetrode equation' S:m,,s = 1.5 R In Mfi + 1.5 R In (T/K) + R(2.5 + In V (271R)? * hP3-Ni4) (4) where R is the gas constant 8.3143 J K-' mol-' Miis the relative atomic mass of the ion h is Planck's constant NA is Avogadro's number Tis the absolute temperature and V is the volume. If the values of the physical constants are introduced the volume of the ideal gas is taken as V= RT/P ' H. Tetrode Ann. Phys. 1912 38,434; 0.Sackur ibid. 1913 40 67. Standard Entropies of Hydration of Ions where P is the standard state pressure of 0.1 MPa and the temperature is set at 298.15 K then equation (4) becomes S,~r,,,(298.15K) = 12.4715 In M + 108.85JK-'mol-' (5) The contribution of the 'magnetic' entropy in the case where only the ground electronic level is populated is R In (2j + l) wherej is the multiplic- ity of the ground level.The value ofj is obtained from the NBS Tables of Atomic Energy Levels.8 This is also the source for the energies of the higher levels amp;k in cm-' from which the quantities uk = 1.43879amp;k/(T/K) and qk = (2jk + l)e-"k (both dimensionless) are obtained and hence9 S:lec = Rln cqk + cukqk/cqkl (4) The summation for T = 298.15 K extends over all those levels k that have amp;k values not exceeding 2000 cm- ' since only these contribute significantly to the entropy. If only the k = 1 level is populated then E = 0 uI = 0 q1 = (2j + l) and SiOelec= R In (2jl + l) and it equals the 'magnetic' spin-only contribution.The standard molar entropy of the gaseous monoatomic ion is thus Sy(g) = S:rans+ SiOelec,where the latter is obtained from equation (6) and where Szrans is obtained from equation (5). The accuracy of the values of Sp(g) of the monoatomic ions is very high (better than 0.01 J K-' mol-') except in the cases of the artificial elements where the mass is arbitrarily chosen as that of a representative long-lived isotope. This arbitrariness intro- duces an uncertainty of less than amp; 0.1 J K-lmol-' in Sp(g) of the ions of these elements. The standard molar entropies of some 115 polyatomic gaseous ions where reported by Loewenschuss and Mar~us.~ Of these only about 50 were reported previously and some of these values required revision in view of the availability of updated input data.The data required for the calculation of Sy(g) of polyatomic ions are the number N of atoms in the ion and their masses the symmetry of the ion the bond lengths and bond angles and the frequencies of all the fundamental vibrations (at least all those having wave numbers below about 1600cm-'). The choice of the ions treated in that review4 depended largely on the availability of the pertinent data in the literature or the ability of the authors to estimate them on the basis of reasonable assumptions. These data form the basis for the calculation of the rotational and vibrational contributions to the entropy that polyatomic ions have in addition to the contributions included in equations (5) and (6).The equations employed for the calculation of the rotational and vibra- tional contributions to the entropy' follow. S:, = R(0.5 In D + 1.5 In (T/K) -In 0)+ 34.90JK-'mol-' (7) C. E. Moore 'Atomic Energy Levels' 1949-1958; W. C. Martin R. Zalubas and L. Hagen 'Atomic Energy Levels of the Rare Earth Elements' 1978 National Bureau of Standards Washington DC. K. S. Pitzer and L. Brewer 'Thermodynamics' McGraw-Hill New York NY 2nd Ed. 1961 Chap- ter 27. Y. Marcus and A. Loewenschuss for a non-linear ion where D is the determinant of the moments of inertia and 0 is the symmetry number. For a linear ion = SEOtlin R( 1 -In y -In (r -y2/90) (8) where y = 0.24254I/(amunm2)-'(T/K)-' and I is the moment of inertia.The vibrational contribution for non-linear ions is 3N-6 s:,~= R 1 u(eu -I)-' -In (1 -e-") (9) I where u = 1.43879(T/K)-'(v/cm-I) and v is the vibrational wave number. For linear ions the summation extends over 3N-5 vibrational frequencies. A relatively large uncertainty arises in the case where internal rotation of a polyatomic part of the ion around a bond to another polyatomic part is possible but generally the uncertainties in the calculated values of S,O(g) are quite low. The uncertainties of the values of Sy(g) of the polyatomic ions from this source4 reported in Table 1 follow the code of being 2 to 10 times the unit of the last digit reported.For further details on the application of these equations and other features of the calculation the original review4 should be consulted. Discussionof the Data. -The list of ions included in Table 1 is determined by the requirement that both S?(g) and S:,,,(g) values be available for each ion entered. This requirement excludes on the one hand ions that react with water to form hydrolyzed species of an indefinite composition so that no SOconv(aq) value can be determined for them via equation (3). Among the monoatomic ions for khich S,O (g) can always be calculated from equations (5)and (6)are excluded for example Nb" Ta5+ Pa" Ti3+ Ti4+ Ge4+ N3-. Some such ions are nevertheless included for the sake of systemization and in the hope that values may eventually be assigned to them or estimated on the basis of correlations; see below.Polyatomic ions that are hydrolyzed by water such as ClSO and PH are also generally excluded. A major reason for the exclusion of many ions that may otherwise have been of interest in the present context is the excessive number N of atoms they contain. Except for some highly symmetrical ions the required 3N- 6 vibrational frequency values are generally not available for N 5 and are unlikely to become available in the future. Some of these frequencies corresponding to bending or torsion modes would have a low wave number and contribute heavily to the vibra- tional entropy of the ion; see equation (9). Ignorance of their values makes a calculation of the total entropy of such ions impossible.In some cases where S,O(g) values are known but no reliable values of Szonv(aq) could be found in the literature the latter may be estimated on the basis of several correlation equations that have been proposed. These provide expressions that relate the conventional standard partial molar entropies of aqueous ions to some other of their properties such as their sizes and charges. Some of these have been formulated in terms of the absolute rather than the conventional partial molar entropies but are recast here in terms of $amp;,,(aq). For monoatomic ions these expressions include the following. Standard Entropies of Hydration of Ions According to Powell and Latimer" -0 -S,,,,,(aq) = 1.5 R In M, + 155 -11.31~~1(ry/nm) + A-2JK-1mol-i (10) where A = 0.20 nm for cations and A = 0.10 nm for anions and rp is the Pauling crystal radius (co-ordinaton number 6) of the ion.This expression was subsequently simplified by Powell" to -0 S,,,,,(aq) = 197 -6.441~~1 -(r,!/nm) + J K-'rnol-l (1 1) where A = 0.13 nm for cations and A = 0.04nm for anions. The first power dependence on z and the reciprocal second power dependence on (the modi- fied) ('were criticized by LaidlerI2 as being inconsistent with the functional dependence expected from the electrostatic contribution to the entropy according to Born's equation." He therefore proposed the expression (valid only for cations having a rare gas electronic configuration) -0 Slco,,(aq) = 1.5R In M, + 43 + 232 -4.86~~(r~/nm)-'JK-~rnol-l(12) where r," is the univalent Pauling radius of the cation.A rebuttal of this criticism led Scott and Hugus14 to their modification of of the original Powell and Latimer equation" -0 S,,,,,(aq) = 1.5R In M, + 153 + 232 -13.47zI(r~/nm)+ 0.20-2JK-'mol-' which was tested for cations only. For polyatomic ions that are oxyanions several other expressions were proposed. These include that by Connick and P~well'~ -0 S,,,,,(aq) = 182 -1952 + 54n,JK-'mol-' (14) where no is the number of oxygen atoms in the anion. This expression is limited to anions having a central atom surrounded by oxygen atoms. When there are two central atoms (as in N,O:- S,O:- or Cr,O?- for example) then each half is treated according to equation (14) the result is doubled and -75 J K-' mol-' is added to account for the 'dimerization'.An alternative expression was proposed by Cobble,16 to deal with any kind of oxyanion -0 S,,,,,(aq) = 1.5R In M, + 276 -33.9~~,~f;(r,-,/nrn)-~JK-~mol-~(15) where ripe is the distance between the centers of the central atom and the peripheral oxygen atoms andA is a structural factor. This factor equals 0.74 for mono- and di-negative tetrahedral and pyramidal anions 0.83 for tri- negative tetrahedral ones 0.68 for plane triangular ones 0.87 for bent and lo R. E. Powell and W. M. Latimer J. Chem. Phys. 1951 19 1139. I' R. E. Powell J. Phys. Chem. 1954 58 528. I' K. J. Laidler Can. J. Chem. 1956 34 1107. l3 M. Born Z. Phys. 1920 1 45. I4 P. C. Scott and Z.Z. Hugus jun. J. Chem. Phys. 1957 27 1421; see also K. J. Laidler ibid. p. 1423. l5 R. E. Connick and R. E. Powell J. Chem. Phys. 1953 21 2206. l6 J. W. Cobble J. Chem. Phys. 1953 21 1443. Y. Marcus and A. Loewenschuss 0.96 for linear triatomic ones 1.34 for diatomic ones and 0.77 for anions of lsquo;complex shapersquo;. The latter are lsquo;dimericrsquo; anions such as N,O:- Cr,O:- and C20i-.Another expression was proposed by Couture and Laidler:rdquo; Szonv(aq) = 1.5R In M, + 168 -231~~1 -45.5zfnamp;rsquo; -(r,-amp;m) + 0.140-1JK-lmo1-l (16) where nb is the number of lsquo;charge-bearing ligandsrsquo; i.e. oxgen atoms to which no hydrogen atoms are attached (the expression being valid also for protonated oxyanions). For polyatomic ions that may be considered as lsquo;complexesrsquo; Cobble18 proposed the expression Sz0,,(aq) = 205 -4l.41zi(J;rsquo;(r,-,/nm)-rsquo; + n,So(H,O l)JK-lsquo;rnol-l (17) wheref is again a structural factor equalling 1 .OO for monoatomic ligands and 1.54 for the ligands NH, CN- NO etc.and nL is the number of water molecules displaced from the hydrated central ion by the ligands taken to equal the number of (monodentate) ligands in the complex for the present purpose. The standard molar entropy of liquid water is3 So(H,O 1) = 69.91J K-rsquo; mol-rsquo;. Finally for oxycations the limited number of cases available still permits the correlationrdquo; Si:on,(aq) = 1.5RIn M -20 -13.3zi(ri-,/nm)-rsquo;JK-rsquo;mol-rsquo; (18) based on the data for the vanadium uranium and transuranium element oxycations. The expressions (10) to (18) are based on liner correlations of ,$rsquo;Eon,(aq) or of this quantity -1.5R In Mrior -23zi or minus both as the case may be obtained from the standard compilations available at the time with the key variable that depends on the charge and size of the ion.The claimed accuracy varied from amp; 5 to 15 JK-rsquo;mol-rsquo; but the differences between values calculated according to the several expressions pertinent to a certain class of ions may exceed these limits manyfold. A re-examination was made of equations (10) to (1 3) for the monoatomic ions and of equations (14) to (16) for the polyatomic ones in the light of the more modern input data now available. For 54 monoatomic ions the standard deviations of the fit with each of the equations was substantially the same f9JK-rsquo;mol-rsquo; so that none proved superior to the others.For ions where $amp;,,(aq) was not found in the NBS Tables3 or other sources the average of the values obtained from these three equations are presented in Table 1. For 25 pyramidal and tetra- hedral oxyanions equation (1 6) gave the smallest standard deviation amp; 6 J K-rsquo; mol-rsquo; but was ony marginally better than the others. It was used for the estimaton of SEconv(aq) for these oxyanions when no other sources of data were available. A. M. Couture and K. J. Laidler Can. J. Chem. 1957 35 202. rsquo;* J. W. Cobble J. Chem. Phys. 1953 21 1446. l9 Y. Marcus and A. Loewenschuss unpublished results 1984. Standard Entropies of Hydration of Ions 91 To anticipate the discussion in Section 5 it is noted that several of the expressions (10) to (18) include the term -23zi that reflects the fact that these expressions originally pertained to the absolute rather than the conven- tional $O(aq) for the ions.This term converts from the former to the latter reversing the procedure used by the original authors. Comments on Specific Ions.-Following are comments on the entries in Table 1 for those cases (marked with an asterisk) where Sy (g) was not taken from Loewenschuss and Marcus4 or ,,,(as) was not taken from Wagman et ~1.~ and for a few additional cases. No. 0,e-. The value of Sy(g) for the electron was taken from the JANAF (1982) supplement2 and converted to the standard state of 0.1 MPa by the addition of R In (0.101325/0.1) = 0.11 JK-'mol-'.The values of SPconv(aq) for the hydrated electron was taken from Jortner and Noyes,20*21 who esti- mated the absolute value as 13.0JK'mol-' to which 22.2JIC'molV' is added for conversion to the conventional value22 (see below) required in Table 1. No. 1. 02-.Oxide anions do not exist as such in dilute aqueous solutions being aquated to hydroxide anions but for the sake of systemization are included here. Equations (10) to (13) yield the mean S,;,,,,,(aq) = -86J K-' mol-I. No. 2 0,. Superoxide anions are also aquated in dilute aqueous solutions and are included for the sake of systemization. If 0; is treated as an oxyanion according to equations (14) and (16) the values 96 and 105 JK-Imol-' respectively are obtained for $l:o,v(aq) if equation (15) is used the value -2OJK-'mol-' results.The latter is less likely to be true since mono- negative ions of similar size (cf. CN-) tend to have positive values of S;,,,"(aq). The mean of the values from equations (14) and (16) is entered in Table 1. No. 3 O;-. Peroxide anions hydrolyze to hydroperoxide HOO- but may have an existence in strongly basic aqueous solutions hence they are included. The values calculated for ~,~,,,,(aq) by equations (14) (1 5) and (I 6) treating 0;-as an oxyanion are -100 -290 and -150JK- 'mol-I respectively and it is difficult to select the most reliable value. In view of the relation between the values for S2-and Sip and the positive $,yonY(aq) of the latter the least negative of the values given above for 0;-is preferred and entered in Table 1.No. 4 H+ and No. 6 H30+.Hydrogen ions hydrated to various extents to H(H2O); have distinct existences in the gas phase (see Section 4) and values of Sio(g) can be assigned to them. The nominal assignment of the conven- tional value of zero to SiOconv(aq) of the aqueous hydrogen ion does not permit a distinction between H+(aq) and H,O+(aq) (or any other of the hydrated hydrogen ion species). When the Sio(g) values for H+ and H,O+ are used 2o R. M. Noyes J. Am. Chem. SOC.,1964 86 971. 2' J. Jortner and R. M. Noyes J. Phys. Chem. 1966 70. 770. 22 B. E. Conway J. Soh. Chem. 1978 7 721. 23 JANAF J. Phys. Chem. Re$ Data 1982 11 695. 92 Y. Marcus and A. Loewenschuss different values of AhydrS~onv of the hydrogen ion necessarily result.For their interpretation consult Section 7. No. 23a At-. An estimate of the standard partial molar entropy of the aqueous astatide anion was given by Kre~tov,~~" S,:o,v(aq) = 126J K-' mol-I. This value is reasonable in view of the values for the other halide anions and is therefore entered in Table 1 together with the calculated S,O(g) value given by equation (5) for the mass of the longest lived isotope 210At. No. 31 S202-.A value of S,~o,v(aq) was reported for this ion in the last edition of Latimer's book2' but was not included in the NBS table^,^ apparently because of its low reliability. The value 125 J K-' mol-' is intermediate between those calculated by equations (14) and (16) 223 and 79 J K-' mol- I respectively.The value calculated by equation (1 5) -21 J K-' mol-' is far off in view of the values for S,O:- S20i- and S,Oi-. A positive value not far from that estimated by Latimer25 is indeed expected and in lieu of a more reliable estimate this value is entered in Table 1. No. 37 Se2-. The selenide ion is hydrolyzed in dilute aqueous solutions to hydroselenide HSe- but should be able to exist in strongly alkaline solu- tions. The value S',:o,v(aq) = 0 was reported for Se2-(aq) by Friedman and Krishnan' without reference to an original source. Another value for this quantity again without reference to the source is -28 J K-' mol-' reported by Ryab~khin.~~~ The values calculated by equations (10) and (1 I) are -28 and -11 J K-' mol-' respectively.In view of the value for S2- however the least negative value namely 0 J K-' mol-' was entered in Table 1. No. 44 TeOi-. The tellurate anion is partly aquated in aqueous solutions to H,TeO:-' but may exist under special circumstance^.^ Equations (14) (15) and (16) yield the values 12 64 and 46JK-'mol-' respectively for s:o,v(aq) of which that given by equation (1 6) comparable with the values for SO;-and SeOt- was adopted in Table 1. No. 46 NO+ and No. 47 NO;. The nitrosyl and nitryl cations are com- pletely aquated in dilute aqueous solutions but may have an existence in very strongly acidic ones. Values of Si:onv (aq) were calculated by equation (1 8) and entered in Table 1. Note that for the ion VO; (as) a value of Ahydr$amp;,vis obtained that is very near to those to NO' and NO presented in Table 1.No. 50 N20i-. The values in Table 1 pertain to the trans-isomer of the nitroxylate anion no entropy value being available for the cis-isomer in the aqueous solution. The value of $Z1,,,(aq) for trans-N,O:-given in Latimer's and adopted by Friedman and Krishnan' and by Bard Jordan and Parsonz6 was entered in Table 1 though it was not endorsed by the NBS table^.^ No. 53 N2Hit. The doubly protonated hydrazinium cation exists in solid salts and in strongly acidic aqueous solutions. The quantity s,:o,.(aq) = 79 J K-' mol-' was reported for it in Latimer's book,*' but was not endorsed 24 (a) G. A. Krestov Radiokhimiya 1962 4 690; (b)A. G. Ryabukhin Zh. Fiz. Khim. 1977 51 968. 25 W. M. Latimer 'Oxidation Potentials' Prentice Hall New York NY 2nd Ed.1952. 26 A. J. Bard J. Jordan and R. Parsons ed. 'Electrode Potentials in Aqueous Solutions' Dekker New York NY 1985 a multi-author revision of ref. 25 published under the auspices of the IUPAC Commissions of Electrochemistry and Electroanalytical Chemistry. Standard Entropies of Hydration of Ions 93 by the NBS table^,^ apparently because of low reliability. Mono- and di-positive ions of comparable size show on increasing the charge a decrease of the standard molar partial aqueous entropy of 92 J K-' mol-' for VO; and V02+ of 69 J K-' mol-' for UO; and UOi+ and 67 J K-' mol-' for three transuranium(V)yl and -(VI)yl cations. On this basis and in view of the value reported for N,H; the value quoted above from Latimer,25 showing a decrease of 72JK-'mol-' on increasing the charge by one unit is quite reasonable.It was therefore entered in Table 1. No. 54 NH30H+. The value of $amp;(aq) given in Latimer's and adopted by Bard Jordan and Parson26 for the hydroxylammonium ion is entered in Table 1 although not endorsed in the NBS table^.^ No. 59 AsO;. The meta-arsenite anion aquates partly in aqueous solu- tions to HiAsO:-' anions27 but it does exist in alkaline solutions2* and in solid salts. A value of $:on,(aq) was assigned to it in the NBS table^,^ which is taken as a further conformation of its existence in solution. The structural data on solid meta-arsenites indicate a polymeric structure as in arsenic(II1) oxide with pyramidal AsO units sharing an oxygen atom.The As4 distance is 0.180 nm (a theoretical calculation gives 0.1866 nm)29 and the 0-As-0 angles are2' 100"and 126'. The mean of these angles is taken for the isolated AsO; anion in the gas phase for the purpose of the calculation of the rotational contribution to Sy(g) according to equation (7). The Raman vibrational frequencies2* 350 533 and 753 cm-' are used with equation (9) for the calculation of the vibrational contribution. However in view of the Raman frequencies reported3' for the isoelectronic SeO and for the anal- ogous BrO anion the freqeuncy of 533 cm-' seems to be too low one in the range of 700 to 900cm-' being expected for the stretching vibration in question. Replacement of this frequency of 533 cm-' by 900 cm-' and vari- ation of the O-As-0 angle in the range from 100" to 126" cause an uncer- tainty of up to f2JK-'mol-' in the caluclated value Sio(g) = 268.2 J K-' mol-' entered in Table 1.The infrared3' and Raman27 spectra of the solid salt NaAsO show more spectral features than correspond to an isolated bent triatomic ion because of the polymeric nature of this salt and are not relevant in the present context. No. 61a SbO+. The entropy of gaseous SbO+ was not included in the re vie^,^ since the value of the standard partial molar entropy of the aqueous antimony1 ion had not been included in the standard corn pi la ti on^.^*^^*^^ Since however there exists the report by Vasil'ev and Shor~khova~~ of Samp;,(aq) = 22 7 J K-' mol-' for this ion obtained from e.m.f.measurements with an 27 T. M. Loehr and R. A. Plane Inorg. Chem. 1968 7 1708. 28 F. Feher and G. Morgenstern Z. Anorg. Chem. 1937 232 169. 29 L. E. Sutton 'Interatomic Distances' Chem. SOC. Spec. Publ. No. 1 I 1958 Suppl. No. 18 1965; see also A. Potier J. Chim. Phys. 1953 50 10 for a calculated value. 30 K. Nakamoto 'Vibration Spectra of Inorganic Compounds' Wiley-Interscience New York NY 3rd Ed. 1977. 3' F. A. Miller and C. H. Wilkins Anal. Chem. 1952,24 1253; F. A. Miller G. L. Carlson F. F. Bentley and J. H. Jones Spectrochim. Acta 1960 16 135. 32 V. P. Vasil'ev and V. I. Shorokhova Eleclrokhimiya 1972 8 185. Y. Marcus and A. Loewenschuss antimony electrode in 0.3 to 2.5 mol dm-3 HClO over the temperature range from 15 to 50 "C,a value of S?(g) was calculated for the present report.The Sb-0 bond length was taken as 0.1807 nm and the vibration frequency as 942cm-' from the work of Tripathi et aZ.33 The total S,"(g) obtained was 231.6 J K-' mol-' with an estimated uncertainty not exceeding 0.6 J K-' mol-'. No. 62 SbOi-. The orthoantimonate anion is aquated in aqueous solutions to HiSbOip',therefore no value of ~~onv(aq) was assigned to it in the NBS table^.^ The entropy of the gaseous anion was calculated, however on the strength of a reported3 Raman spectrum of the isolated tetrahedral species SbOi- and an estimated interatomic distance Sb-0. If the value of S,"(g) is accepted then it is also instructive to estimate $,,,(aq) by equations (14) (15) and (16). The results are -184.-125 and -155JK-'mol-' respec- tively and in view of the results for PO:-and AsOi- the value obtained by equation (1 6) was entered in Table 1. No. 63,Bi3+.No value of SiOconv(aq) is given in the NBS Tables3 for this highly hydrolysed cation which does however exist in moderately acidic solutions. The e.m.f. data of Vasil'ev and Gla~ina~~ and auxiliary data3 permit the estimation of this quantity. The standard e.m.f. at 298.15K for the cell reaction Bi(s) + 3H+(aq)+ (3/2) H,(g) + Bi3+(aq) was found to be 0.3172 V after application of the Debye-Huckel correction to zero ionic strength of the measured results. The temperature coefficient dEo/dT = 1OP4 V K-' was also determined in that From this the (1.30 0.15) standard entropy change for this reaction -12.4J K-' mol-' is obtained and together with the values of So(Bi s) = 56.7 and So(H, g) = 130.7 J K-' mol-' this gives for $;,,,(aq) the value -15 1.8 J K-' mo1-I.Vasil'ev and Gla~ina,~~ however reported -176.6 J K-' mol-' for this quantity from the same data but using auxiliary data from a different source and not specifying the convention for $O(H+ aq) used in their calculation. A later report by Vasil'ev and Ik~nnikov,~~ based on heats of solution data gives an even more negative value for Si:onv of Bi3+(aq) again without specification of the convention and the auxiliary data used. Ryabukhin'" reported for this quantity the value -175.1 JIC'mol-' without giving the source. Kozin et aL3' reported the much less negative value of -25.8 J K-' mol-' based on e.m.f.measurements in solutions containing bromide anions and the Bi+ cation. This value must pertain to some bromo-complexed species of bismuth(III) since it is by far not negative enough for a trivalent though large cation. Entered in Table 1 is the value based on dEo/d Tof Vasil'ev and Gla~ina,~~ the determination of which being the best documented. Nos. 73,(CH3)4N+,74 (C,H,),N+ and 75,(C3H7)4N+. The standard partial molar entropies of these aqueous ions were not tabulated in the NBS table^.^ 33 R. Tripathi S. B. Rai and K. N. Upadhya J. Phys. B 1981 14 441. 34 N. I. Ushanova A. M. Aleksandrovskaya and M. V. Nikonov Zh. Prikl. Spektrosk. 1978 28 356. 35 V. P. Vasil'ev and S. R. Glavina Electrokhimiya 1969 5 413.36 V. P. Vasil'ev and A. A. Ikonnikov Zh. Fiz. Khim. 1971 45 292. 37 L. F. Kozin A. G. Egorova and N. N. Gudeleva Ukr. Khim. Zh. 1982,48 688. Standard Entropies of Hydration of Ions 95 A report by Le~ien,~ for tetramethylammonium samp;(aq) = 209 JK-lmol-' based on solubility measurements and heats of solution is in substantial agreement with the subsequent report by Johnson and Martin.39 These authors reported values for both S,",,,(aq) and S,"(g) for all three ions. Loewenschuss and Marcus4 discussed the problem of the free rotation of the methyl groups in the first named ion around the N-C bonds and concluded that the tetramethylammonium ions should be considered as rigid. In this case the symmetry is not Td and the value of R In 12 presumably subtracted according to equation (7) for Td symmetry by Johnson and Martin39 (without giving any details of the calculation in their paper) should be added back.This brings their S?(g) value to within 3 J K-' mol-' of the value4 entered in Table 1 for (CH3)4N+. Corrections for hindered rotation around the N-C bonds lowering the symmetry were then applied4 to Johnson and Martin's values39 for the other two ions to give the values entered for them in Table 1. The values given by these authors for conv(aq) were incorporated into the Table without change no further estimates being available. Their uncertain- ties were given as f3 *4 and * 5 J K-' mol-' for the tetra-methyl- -ethyl- and -propyl-ammoium ions respectively. No. 79,SnFi-.The estimate of SPconv(aq) = 0 for the hexafluorostannate(1V) anion given in Latimer's book2' was adopted by Friedman and Krishnan5 and by Bard Jordan and Parsons,26 but was not endorsed by the NBS Table.3 A much higher positive value 220 J K-' mol-' is obtained by the application of equation (1 7) and is more in line with values presented for other dinegative hexahalometallate anions e.g. SiFi- and PtCl2- in the NBS Tables. This calculated value is therefore preferred and entered in Table 1. No. 100 Ag(NH,),f. The value of S,O(g) of the well established silver diammine complex was inadvertently omitted from the review of Loewen- schuss and Marcus4 and is added here. The ion is considered as a rigid linear species without free rotation of the NH groups around the Ag-N bonds since a value of the torsion frequency of these groups around this bond was reported? There are four skeletal vibrations two Ag-N stretching vibra- tions at 400 and 476 cm-' and two N-Ag-N bending vibrations at 21 1 and 221 cm-'.The vibrations associated with the NH,-groups are one torsion frequency at 265 cm-' two doubly degenerate rocking vibrations at 648 and 653 cm-' two H-N-H bending vibrations at 1283 and 1300cm-' and additionally two non-degenerate and four doubly degenerate further vibra- tions at frequencies 1600cm-' that are immaterial for the present pur- poses. The Ag-N bond length was given as 0.188 nm and the N-H one as 0.103 the NH group being regular-tetrahedrally4' bonded to the silver atom the symmetry of the ion being and the symmetry number t~ = 6.With the masses of the atoms the NH being considered as a 'heavy nitrogen' atom at its center of gravity these are all the data required for the calculation of S,O(g) 38 B. J. Levien Aust. J. Chem. 1965 18 1161. 39 D. A. Johnson and J. F. Martin J. Chem. SOC.,Dalton Trans. 1973 1585. M. G. Miles J. H. Patterson C. W. Hobbs M. J. Hopper J. Overend and R. S. Tobias Inorg. Chern. 1968 7 1721; A. L. Geddes and G. L. Bottger ibid. 1969 8 802. 96 Y. Marcus and A. Loewenschuss at 298.15 K. The contributions are Siamp; = 170.5 Si:ot= 33.6 and amp;", = 37.1 to give a total S:(g) = 241.1 JK-'mol-'. With this value however the standard partial molar entropy of the aqueous ion reported in the NBS Tables,' s~o,v(aq) = 245.2 J K-' mol-I seems to be much too high since it leads to an exceptional positive value for the standard molar entropy of hydration of this ion.It is also much higher than the values reported there' for Pt(NH,):+ and Co(NH,):+. The value calculated according to equation (17) 201 J K-' mol-' is more reasonable in this respect. No. 106,Co". The energy levels of Co'+(g) are not listed in the compilation of Moore' and are instead taken from that of Shugar and Corliss4' for cobalt at various stages of ionization. No. 113 Pd2+. The value szonv(aq) = -184 J K-' mol-' reported in the NBS Tables3 is considerably more negative than expected compare the value for Ni2+(aq) -128.9 J K-' mol-I and the discusson concerning Pt2+(aq) below. An estimate of the entropy of hydration AhydrSi:onv = -176 JK-'mol-' by Watt et aZ.,42 with the comment that the standard partial molar entropy of Pd2+(aq) is unusually high it comes out to be about zero according to this value of the entropy of hydration and S:(g) leans too much on the other side.Since the crystal ionic radius of Pd2+ is about the same as that of Pt2+ (see below) a value of si",,,(aq) is expected. No. 115 Pd(NH,);'. A value of Siamp;(aq) of the palladium(I1) tetraammine cation was reported neither in the NBS Tables' nor elsewhere although a value for the corresponding platinum complex cation was reported there. Application of equation (17) to Pd(NH,):+ yielded the value 307 J K-Imol-' (with the Pd-N bond length reported4) which is much higher than the value 44.8 J K-' mol-' reported3 for Pt(NH,);+ the calculation by equation (17) for the latter yielding the value 236JK-'mol-'.Since there are no other means to evaluate the standard partial molar entropies of these ions the situation for these ammine complexes as also for the silver one (see above) remains unsatisfactory. No. 117 Pt2+. The energy levels of Pt2+ are not listed in the compilation of Moore,' and no other source for them could be found. They were therefore approximated by those of the isoelectronic Ir+ (g).43 The standard partial molar entropy of the aqueous platinum(I1) cation is not reported in the NBS Tables,' although that of the analogous palladium(I1) cation is. Estimates by equations (lo) (ll) and (13) yield the values -68 -95 and -79 J K-' mol-' respectively for Pt"(aq) whereas the value reported3 for Pd2+(aq) -184 J K-' mol-' is much more negative.The calculated values for Pt2+ are in line with those reported' for divalent transition metal ions of similar size (but lower mass) Mn2+(aq) -73.6 Zn2+(aq) -112.1 Cu2+(aq) -99.6JK-'mol-'. They seem to be more nearly correct than the very negative value of Pd2+(aq) reported in the NBS table^.^ The value obtained by equation (1 3) -79 J K-' mol-I is near the mean of the calculated values and is adopted in Table 1. 4' J. Shugar and C. Corliss J. Phys. Chern. Ref-Data 1981 10 1097. 42 G. D. Watt D. Eatough R. M. Izatt and J. J. Christensen Proc. Utah Acad. Sci. 1965 42 298. 43 Th. A. M. Van Kleef and B. C. Metsch Physica C 1978 95,251.Standard Entropies of Hydration of Ions 97 No. 126 Re-. The rhenide anion Re- was said to be obtained by strong reduction of the perrhenate anion inacidic solution and was assigned a value of s:,,,(aq) in the NBS table^.^ However its existence as a hydrated mono- valent anion was questioned and it was suggested that the species is actually a hydride.44 The value of s",,,(aq) = 230 J K-' mol-' assigned to the anion Re-(aq) is much too positive compared for instance with that of I-(aq) 111.3J K-' mol-'. It seems not to be acceptable since it leads to an excep- tional positive value of AhydrSi:o, of this ion. Nos. 129 Cr2+,and 130 Cr3+. No values of sgonv(aq) were given for the aqueous chromium(I1) and chromium(II1) cations in the NBS table^.^ Vasil'ev et al.45 measured the solubility of NH4Cr(S04)2 12H,O (ammon- ium chromium alum) in water and its heat of solution at 298.15K and estimated the activity coefficient and the water activity of the saturated solution.From these data they derived sgonv(aq) = -269 7 J K-' mol-' for Cr'+(aq). A much less negative value -215 J K-'mol-' was reported by Ryab~khin,~~~ without giving the sourse of the data. An estimate was given also in Latimer's book2' and was adopted by Bard Jordan and Parsons,26 -293 J K-' mol-'. The values calculated by equations (1 0) (1 l) and (1 3) and -290 -318 and -328 J K-'mol-' respectively. For Cr*+(aq) there is no experimental value available and the value reported in the compi- lation~,~'.~~ -100 JK-' mol-I and those obtained from equation (lo) (1 I) and (13) -76 -85 and -86 respectively must form the basis of the choice.The mean of these estimates -82 J K-' mol-' for Cr"(aq) and the experimental value -269 J K-' mol-' for Cr3+(aq) were entered in Table 1. Nus. 135 V2+,and 136 V3+. No values of sgonv(aq) were given for the aqueous vanadium(I1) and vanadium(II1) cations in the NBS table^,^ nor were they given in other reference compilations. The values calculated by equation (lo) (1 l) and (13) are -69 -75 and -77 JK-'mol-' for V2+(aq) and -286 -322 and -314J K-' mol-' for V3+(aq) respectively. For the lack of a better criterion the means of these values were adopted in Table 1. No. 140 VOi-. No value of $Oo,,(aq) was reported in the NBS Tables3 for the orthovanadate(v) anion nor was it in other reference compilations.The value given for the misprinted 'VO; ' by Friedman and Krishnan pertains perhaps to the mono-negative metavanadate anion VO, but not to the tri-negative orthovanadate anion VOi-. The values calculated for the latter anion by equations (14) (15) and (16) are -184 -160 and -172 JK-'rnol-' respectively. In view of the value for AsO:- which is approximately of the same size (slightly smaller); the mean of the three values was adopted for Table 1. Nus. 142 Zr4+,and 143 Hf4+. These highly hydrolysable cations are stable in very acidic solutions (above say 1 moldmP3 acid) but no values of Szonv(aq)were assigned to them in the NBS Tables3 or in other reference 44 F.A. Cotton and G. Wilkinson 'Advanced Inorganic Chemistry' Interscience-Wiley New York NY 1st Ed. 1962. In later editions up to the 4th 1983 the -1 oxidation state of rhenium was illustrated only by a carbonyl complex never by a 'rhenide' anion. 45 V. P. Vasil'ev V. N. Vasil'eva 0. G. Raskova and V. A. Medvedev Zh. Neorg. Khim. 1980 25 1549. 98 Y. Marcus and A. Loewenschuss compilations. However Vasil'ev and L~tkin~~ studied the thermodynamics of their solutions calorimetrically and obtained values for the standard partial molar entropies of Zr4+(aq) and Hf4+(aq) that are entered in Table 1. Nos. 147 Yb2+,and 157 Sm2+. These strongly reducing ions in aqueous solutions are less stable than Eu2+(aq) though capable of existence under special circumstances.Whereas the latter cation was assigned a value of sgonv(aq) in the NBS table^,^ the former two cations were not. However estimates were given to them in Latimer's that were adopted by Bard Jordan and Parsons,26 and these are entered in Table 1 since they are of the expected magnitude for ions of their relatively large size see Sr?+(aq) and Ba2+ (as). No. 159,Pm3+. The aqueous cation of this highly radioactive element was not assigned a value of sgonv(aq) in the NBS table^.^ Tremaine and G01dman~~ reported the standard partial molar entropy of Pm3+(aq). Their values for the other lanthanide cations yield sgonv(aq) less negative by 3.4J K-' mo1-' on the average than the values reported in the NBS table^,^ and this amount was therefore subtracted to yield the value entered in Table 1 that fits in well between the values for Nd3+(aq) and Sm3+(aq).Nos. 165 Bk3+,167 Cm3+,168 Am3+,171 Pu3+,175 Np3+,179 U3+,and 184 Ac3+.No values of sgon,(aq) for the trivalent actinide cations were reported in the NBS table^.^ Some of them were reported by Tremaine and G01dman~~ and some others by Bard Jordan and Parsons.26 Still other estimates were made by Lebede~,~* which are in reasonable agreement with the other ones. The differences are generally within a 1.5 JK-lmol-' and the means of the values were adopted for Table 1. No. 166,Bk4+.No value of s:onv(aq) was reported in the NBS Tables3 for the aqueous Bk" ion which though highly hydrolizable exists in strongly acid and oxidizing media. A new determination of the temperature dependence of the redox potential of the Bk"/Bk"' couple by Simakin et aZ.49 yielded a value for the entropy change of the reaction Bk3+(aq) + H+(aq) e Bk4+(aq) + +H2(g).Together with s:onv(aq) adopted here for Bk3+(aq) and the entropy of H2(g) this yields -395 4 for the standard partial molar entropy of Bk4+(aq) presented in Table 1. Nos. 172 Pu4+,and 176 Np4+.No values of sgonV(aq) of these ions were presented in the NBS table^,^ but estimates are given in Latimer's A recent re-evaluation was presented in the compilation of Bard Jordan and Parsons,26 and these values are entered in Table 1. No. 180 U4+.No value of was presented for the highly hydroliz- able uranium(1V) cation in the NBS table^,^ but an estimate appeared in Latimer's A later estimate based on the redox potential of the Uv'/U1vcouple was presented by Sobkolo~ski.~~ The recent compilation by 46 V.P. Vasil'ev and A. I. Lytkin Zh. Neorg. Khim. 1976 21 2610. 47 P. R. Tremaine and S. Goldman J. Phys. Chem. 1978 82 2317. 48 I. A. Lebedev Radiokhimiya 1978 20 641. 49 G. A. Simakin A. A. Baranov V. N. Kosyakov G. A. Timofeev E. A. Erin and I. A. Lebedev Radiokhimiya 1977 19 373. 5o J. Sobkolowski J. Inorg. Nucl. Chem. 1962 23 81. Standard Entropies of Hydration of Ions Bard Jordan and Parsons26 presents the re-evaluated quantity reported in Table 1. Nos. 169 AmO; 170 AmO:+ 173 PuO; 174 PuO;' 177 NpO and 178 NpO:' .No value of samp;(aq) were reported in the NBS Tables3 for these actinide(v)-yl and (V1)-yl ions contrary to the cases of UO; and UO;'.The values reported by Bard Jordan and Parsons,26 that were adopted also by Lebedev" were entered in Table 1. 3 Entropies of Hydration at Elevated Temperatures Standard molar entropies of hydration of ions at temperatures other than 298.15 K are obtained by equation (2) from their standard molar entropies in the gas phase SF(g) and standard partial molar entropies in the aqueous phase $O(aq) at these temperatures. The subscript conv in equations (2) is dropped here since the discussion pertains to both the conventional and the absolute (see Section 5) entropies of hydration. The standard molar entropies of gaseous ions at elevated temperatures are obtained as before for 298.15 K for monoatomic ions from equations (4) and (6) and for polyatomic ions from additionally equations (7) or (8) and (9).Note that equation (4) contains T in the volume V so that the total tem- perature dependence of Siamp; is 2.5R In (T/K). The translational and rotational contributions together are then S,zans+rot(g = S17rans+rot(g, T) 298.15K) + 33.26 In (T/298.15)JK-'mol-' (19) For monoatomic ions with inert gas electron configurations and also for those with an inert electron pair beyond that equation (19) is sufficient for the calculation of Sy(g T).For other ions where electronic and vibrational contributions must be included the calculations are more involved. Within the range of the existence of water as a liquid i.e. up to its critical tem- perature of 647.35K electronic levels need to be considered only up to 4500 cm-' and vibrational levels only up to 3500cm-'.Beyond these energies their contribution is negligible. Contributions to the entropy from vibrations up to 2000cm-' at temperatures up to 573 K have been tab~lated.~ For all the ions included in Table 1 the required energy level (vibration frequency) data are available in the sources employed for the calculation of Sy(g) at 298.15 K for monoatomic' and for polyatomic ions.4 For a few polyatomic ions Krestovs2 reported calculated values of SF(g) at 273.15,298.15,400,500 and 1000 K. The standard partial molar entropies of the aqueous ions are less readily available. There is no single self-consistent source for the entropies of electrolytes at elevated temperatures comparable to the NBS Tables3 for 298.15K.Values of s?(aq) for thirty three ions at 333.15 373.15 and 423.15 K were reported by Criss and Cobbles3 (as absolute ionic partial molar entropies). Entropies of hydration were reported by Lebed' and Aleksan- " I. A. Lebedev Radiokhimiya 1981 23 12. 52 G. A. Krestov Zh. Neorg. Khim. 1968 42 866. s3 C. M. Criss and J. W. Cobble J. Am. Chem. SOC.,1964,86 5385 and 5390. Y. Marcus and A. Loewenschuss Table 2 Parameters for equation (20) aj(T)/JK-I mol-' = ajo+ ajlT + aj2T2 and bj(T) = bjo + bj,T + b, T2,valid to 423 K (with possibly larger errors to 573K). j 40 ajI I 0 3 ~ ~ ~bj I 03bj I 06bj2 monoatomic cations -158 0.508 0.068 1.478 -1.53 -0.20 halide anions OH- 200 0.629 -0.130 0.985 0.06 -0.13 oxyanions 536 -1.941 0.469 -0.843 6.55 -1.11 protonated oxy- 508 -1.715 0.047 -2.732 13.36 -2.84 anions For the calculation of conventional slmdard partial molar ionic entroies add zi (133.5 -0.03634T -3.713.10-5T2)J K-lrnolK' where zi refers to the charge of the actual ion i irrespective of the class j to which it belongs.drovs4 for the chlorides bromides iodides and hydroxides of Li+ Na+ NH; ME2+ Ba2+ and Co2+ except for Co(OH), at evenly spaced tem- peratures between 273.1 5 and 353.15 K (for some up to 393.15 K). To overcome the lack of specific information for given ions at given tem- peratures several generalizations have been proposed that permit the cal- culation of p(aq T)from s:(aq T,),where T is a reference temperature generally 298.15 K.The best known and simplest of these is the one due to Criss and Cobble,53 known as the 'correspondence plot' $yix(aq T) = aj(~)+ bJ(T)SlS(aqTr) (20) where a and bJare temperature-dependent parameters that fortunately are not ion-specific but common to groups of ions j. The values of aj and bJwere reported at 333.15 373.15 and 423.15K in addition to the values a,(298.15K) = 0 and bJ(298.15K) = 1 for all j. Estimates were also given for 473.15 523.15 and 573.1 5 K. To a sufficiently good approximation the values of these parameters are quadratic functions of T (not given in the original paperss3) which are shown in Table 2. Conversion from the absolute ionic entropies given by equation (20) to conventional ones is by means of the addition of the values of -Z,$;,,~(H+ aq T) values given by Criss and Cobble,53 and also shown as a quadratic function of T in Table 2.As seen in Table 2 for each of the four groups of ions six parameters must be specified in order to calculate s,;,,(aq T).This situation was simplified by Khodakov~kii.~' If it is true as he maintained that the conventional partial molar heat capacity of aqueous ions is proportional to the absolute tem- perature then since it follows that -0 Siconv(aq T) = S:onv(aq Tr) + t(T/Tr) -11c;jconv(aq (22) Tabulated values of the conventional partial molar ionic entropies and heat capacities at T = 298.15K yield via equation (22) values for any other temperature. The former quantities are listed in Table 1 and the latter may be 54 V.I. Lebed' and V. V. Aleksandrov Efektrokhimiya 1965 1 1359. 55 I. L. Khodakovskii Geokhimiya 1969 1 57. Standard Entropies of Hydration of Ions 101 found e.g. in the work of Helgeson et al.rdquo; for some 45 ions. Khodakovskiiss proceeded however to give an empirical expression for the heat capacity on the convention that Ciconv(H+, aq) = 01 -0 Cplconv(aq Tr = a; -~IZ,I-+Sw(aq ~r) (23) Here again as and arsquo; are parameters specific for classes of ions but not to individual ones and they are temperature-independent. For cations and anions that are not oxyanions a; = 212.5 for oxyanions a; = 334.7; for cations arsquo; = 124.7 and for all anions d = 31 1.3 all values in JK-lmol-rsquo;. Equations (22) and (23) thus permit the calculation of S,tonv(aq T)with only two parameters and the known value at T,.The authorss6 presented a correspondence plot of Slyonv(aq 373 K) vs. SPconv(aq,298 K) demonstrating the validity of equations (22) and (23). If the molar entropy or the Gibbs free energy of hydration of a salt is known at some temperature Trsquo; beyond the reference temperature T = 298.15 K then the method of Cobble and Murrays7 is applicable. These authors defined a radius parameter R,and a salt parameter C,for a given salt by the solution of the following two simultaneous equations This makes use of the Born eq~ation,rsquo;~ e being the charge of the electron E, the permittivity of vacuum and E the relative permittivity of water at the specified temperature where also its temperature derivative is to be taken.These authors have shown that at temperatures beyond say 400K the electrostatic interactions described by the Born equation become much more important than other structural contributions to the hydration entropy. Once these two temperature-independent parameters have been evaluated the standard molar entropy of hydration of a salt at any temperature T is given by AhydrSo(T) = (NAe2/8?t~0R,)E(T)-2(d~/dT)p-C (26) No actual application of this set of equations for the calculation of entropies of hydration was presented by the however. In a recent paper by Helgeson et aZ.56 a multiparameter equation for the calculation of standard partial molar ionic entropies was presented. For temperatures up to T = 373.15 K i.e.as long as the pressure equals the reference pressure (1 atm = 0.101325MPa) -0 Stconv(aq = sEonv(aq ~r+ c:conv(aq Tr) ln (T/Tr) + C In (T -o,)/(~r-o,)l+ (NA~~z,~/~~amp;O~,~Y(T) -Y(Tr)I (27) 56 H. C. Helgeson D. H. Kirkham and C. G. Flowers Am. J. Sci. 1981 281 1249 (the tabular material referred to is on pp. 1414 1434 and 1435). 57 J. W. Cobble and R. C. Murray jun. Furaduy Disc. Chem. Soc. 1978 64,144. 102 Y. Marcus and A. Loewenschuss where T = 298.15 K as before Ci and Oi are ion-specific temperature- independent parameters rieff= rc + 0.094qnm for cations and = rc for anions (rcis the crystal ionic radius) and Y = E-~(~E/~T)~. The last term is recognized as arising from the electrostatic interactions according to the Born equation.Values of Ciand Oi were tab~lated,~~ as were those of the coefficient of the difference of the Yrsquo;sin the Born term and the values of camp;,nv(aq c). At temperatures above 373.15 K where the (saturation vapour) pressure of water is higher than atomospheric a further term involving the pressure and more ion-specific parameters must be added to equation (27). The usefulness of this approach seems to be impaired by the proliferation of ion-specific parameters and auxiliary data cf,conv(aq Tr),E( T),and (d~/dT)~. 4 Entropies of Hydration of Ions in the Gas Phase The application of high pressure mass spectrometry has made possible measurements on the extent of the reaction X(H,O):,-I (g) + H*O(g) +X(H,O):,(g) (28) as a function of the temperature.Gas phase ions were produced and electro- statically directed into a reaction chamber containing a known pressure (100 to 2500 Pa) of water vapour. The relative ion-cluster concentrations were determined by mass spectrometry. From the calculated equilibrium constants and their temperature dependence the Gibbs free energy enthalpy and entropy for the above reaction were evaluated. Much of the experimental work reviewed below has been carried out by the groups of Kebarle in Canada and of Castleman in the U.S.A. The ion for which the successive stages of the hydration process were most extensively studied was H+. The primary ions produced in the ionization chamber are OH+ and H20+,which react very rapidly with another water molecule to form H30+.The thermodynamics of the first hydration stage could therefore not be studied.As the reaction is exothermic equilibrium measurements can be performed only if a buffer gas is also introduced to provide third body deactivation. The first such study was reported by Kebarle et al.58Improved apparatus involving the use of a pulsed electron beam for ionization facilitated the updating of the values for the first few hydration stages.59 The predominant contribution to the negative value of the entropy change of reaction (28) may be related to the loss of translational entropy. Some compensation for the rotational entropy loss occurs with the gain of vibrational entropy. An additional entropy gain may be assigned to a certain non-localization of the charge in the cluster formed.The gas-phase proton-water clusters were also studied by the group of Field. In his first results@ no entropy values for the first hydration stages could be obtained and those for the higher stages were essentially in agree- rsquo;*P. Kebarle s. K. Searls A. Zolla J. Scarborough and M. Arshadi J. Am. Chem. SOC.,1967 89 6393. 59 A. J. Cunningham J. D. Payzant and P. Kebarle J. Am. Chem. SOC.,1972 94 7627; E. P. Grimsrud and P. Kebarle ibid. 1973 95,7939. F. H. Field J. Am. Chem. SOC.,1969 91 2827. Standard Entropies of Hydration of Ions 103 ment with those of Kebarle et al.58959 In later work Field et a1." reported very different values for A1,2S0and AZ,S0 of reaction (28) the former even attaining a positive value.In the second of the two reports,61 for which the buffer gas was changed from methane to propane yet another set of values was given still in disagreement with the results of Kebarle et aZ.58*59 The value of A1,2S0was then rechecked62 and found to be in good agreement with the previous results. However when the experiments were repeated with the pulsed beam method,63 that used by Kebarle et al.,58,59 rather than with a continuous beam as in their previous investigations,6M2 acceptable agree- ment with the results of Kebarle et al. was obtained and it was conceded that in the continuous beam studies true equilibrium was not obtained for the first few hydration stages. Recently the group of Kebarlea repeated the exper- iments using updated instrumentation with the view of providing a set of 'best values' of the thermodynamic functions of proton-water clusters and these are also accepted by us as the prevailing values for the entropies of the various stages of reaction (28).It must also be concluded that the question whether measurements were conducted under equilibrium conditions is of crucial importance for the reliability of the thermodynamic data of ion-water cluster systems. All the values reported in the studies discussed above are presented in Table 3. Also included there are values by Godnev et al.,65 calculated on the basis of mass-spectrometric data. If the thermodynamic data are to be related to structural assumptions a start of a new hydration shell would imply a drop in the absolute value of An-,,nSobeyond say n = 4 due to the larger freedom of motion of the outer molecules.An increase of the absolute value of A4,5Sowould be interpreted as evidence for crowding of the first hydration shell. Although the entropy data do not show a clear-cut break after the fourth hydration stage when taken in conjunction with the enthalpy and Gibbs free energy data a preference for the co-ordination number four is indeed indicated.@ There are several other positive ions for which the thermodynamics of hydration in the gas phase was investigated. The alkali metal ion-water clusters were studied by Dzidic et a1.66 and the potassium ion clusters by Searls et al.67 A value of A4,5Sofor the sodium ion-water cluster was reported by Tang et a1.68 and the ammonium ion-water clusters were studied by Payzant et aE.69" and more recently by Me~t-Ner,~~~ with substantial differences between the two sets of values.The water clusters of Cu+ and Ag+ were studied by Holland et al.,70 but equilibrium states for the first two hydration 6' D. P. Beggs and F. H. Field J. Am. Chem. SOC.,1971 93 1567 and 1576. 62 S. L. Bennet and F. H. Field J. Am. Chem. Soc. 1972 94 5186. 63 M. Meot-Ner and F. H. Field J. Am. Chem. SOC.,1977 99 998. 61 Y. K. Lau S. Ikuta and P. Kebarle J. Am. Chem. SOC.,1982 104 1462. 65 I. N. Godnev N. I. Ushanova A. M. Aleksandrovskaya and T. V. Dmitrieva Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 1975 18 554. 66 1. Dzidic and P. Kebarle J. Phys.Chem. 1970 74 1466. 67 S. K. Searls and P. Kebarle Can. J. Chem. 1969 47 2619. I. N. Tang and A. W. Castleman jun. J. Chem. fhys. 1972 57 3638. 69 (a) J. D. Payzant A. J. Cunningham and P. Kebarle Can. J. Chem. 1973 51 3242; (b)M. Meot-Ner J. Am. Chem. Soc. 1984 106 1265. 70 P. M. Holland and A. W. Castleman jun. J. Chem. Phys. 1982 76 4195. 104 Y. Marcus and A. Loewenschuss stages of the former ion could not be attained. It was also argued that for both of these