4 standard entropies of hydration of ions -...
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4 Standard Entropies of Hydration of Ions
By Y. MARCUS and A. LOEWENSCHUSS
Department of Inorganic and Analytical Chemistry, The Hebrew University of Jerusalem, 9 I904 Jerusalem, 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) - X'(aq) (1)
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, J . Phys. Chem., 1985, 81, in the press.
A. Ben-Naim and Y . Marcus, J . Chem. Phys., 1984, 81, 2016.
81
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82 Y. Marcus and A . Loewenschuss
the present topic, in a compilation by Wagman et aL3 and in a review by Loewenschuss and M a r ~ u s . ~ 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
(2) -0 A h y d r C o n v = &o,,(as) - SP<d
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,S0 value to a value of $&(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,6 hence 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 r e p ~ r t e d . ~
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.
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Standard Entropies of Hydration of Ions 83
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 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 23a 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
Ion e- O2 - O2 - 0;- H+ OH- H30+ HOT F- HF; c1- c10 - ClO, ClO; c l o d Br- Bry BrO- BrO; BrO; I - 1; I O j 10; At - S2 - s; - s: ~
so: ~
so;- s20:- S,O,z- s,o; ~-
s20;- s,o; - HS - HSO; HSO, Se2- SeO:- SeO:- HSe- HSeO; HSeO; TeO: - TeOi-
NO+ NO:
N;
NO; NO;
* * * * *
*
*
* *
*
*
*
s? (g) 20.98
143.32 203.8 199.6 108.84 172.3 192.8 228.6 145.59 21 1.3 154.40 215.7 257.0 264.3 263.0 163.57 326.6 227.2 278.7 282.1 169.36 334.7 288.2 297.0 167.47 152.14 223.1 285.4 264.3 263.6 291.1 319. 337.3 341. 356. 186.2 266.8 283.0 163.42 284.0 28 1.2 203.8 283.0 295.8 294.5 295.7 212.2 198.4 214.1 236.3
si%v (aq) 35.2
- 86. 100.
- 100. 0.00
0.00 - 10.7
23.8
92.5 56.5 42.
101.3 162.3 184.0 82.4
215.5 42.
161.7 199.6 111.3 239.3 118.4 222. 126. - 14.6
- 13.8
28.5 66.1
18.8 67. 92.
125. 244.3 257.3
66. 139.7 131.8
0. 13. 54.0 80.
135.1 149.4 13.4 46.
107.9
- 29.
- 103. - 93.
123.0 245.2 146.6
~ h y d r $on,
14.2 - 229. - 104. - 300. - 108.84 - 183.0 - 192.8 - 204.8 - 159.4 - 118.8 - 96.9 - 174. - 155.7 - 102.0 - 79.0 -81.2 - 111.0 - 185. - 117.0 - 82.5 - 58.1 - 95.4 - 169.8 - 75. -41. - 166.7 - 194.6 - 219.3 - 293. - 244.8 - 224. - 227. -212. - 97. - 99. - 120. - 127.1 - 151.2 - 163. -271. - 227.2 - 124. - 147.9 - 146.4 -281.1 - 250. - 104.3 -301. - 307. - 113.3 - 98.6
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No.
50 51 52 53 54 55 56 57 58 59 60 61 61a 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 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 I04
Ion N20i- * NH:
N2Hi+ * NH,OH+ * N2H:
Po: - P,O': - HPOi- H, PO; AsO; * AsOj-
SbO+ * Sb0:- * Bi3+ *
HASO:-
co: - c,o: - HCO; HCO, CH3CO; CN- CNO- SCN- CH3NH: (CH3),N+ * (C,H,)4N+ * (C3H7)4N+ * SiFi- Sn2 +
Sn4 +
SnFz- * Pb2+ BO; BH; BF;
A10; Al(0H); Ga3+
T1+
Zn2+ Cd2+ Hg2+ Hg:+ c u + cu2 +
Ag+ AgC1; Ag(NH3): * Ag(CN);
B(OH); ~ 1 3 +
1 ~ 3 +
~ 1 3 +
AuCl; Au(CN); Ni2-
SP (g) 256.9 186.3 230.5 225.2 235.4 266.4 342.8 283.0 280.7 268.2 282.9 302.9 23 1.6 298.5 175.90 246.1 295.1 238.2 257.9 278.7 196.7 218.9 232.5 327.7 331.9 483. 641 * 309.9 168.52 168.52 354.0 175.49 215.8 187.7 267.9 270.5 149.98 229.5 293. 161.86 168.11 175.32 175.32 161.06 167.84 175.09 273.0 161.05 175.95 167.44 290.7 241.1 284.5 363.8 284.5 178.05
Y. Marcus and A . Loewenschuss
c o n " (aq) 28. 96.9
151. 79.
155. - 222. - 117. - 33.5
92.5 40.6
- 168.9 - 1.7 22.
- 155. - 151.8 - 43.5
45.6 92. 98.4 86.6 94.1
106.7 144.3 142.7 210. 283. 336. 122.2 - 17. - 117.
220.
- 37.2 10.5
110.5 180. 101.2
- 321.7 - 8.8 24.6
- 331. - 151.
125.5 - 192. - 112.1 - 73.2 - 32.2
65.5 40.6
72.68 - 99.6
231.4 245.2 172. 266.9 172.
- 128.9
Ahydr sp0fl" - 229. - 89.4 - 80. - 146. - 80. - 488.0 - 460. - 3 16.5 - 188.2 - 227.6 -451.8 - 304.6 - 210. - 454. - 327.7 - 289.6 - 249.5 - 146. - 159.5 - 192.1 - 102.6 - 112.2 - 88.3 - 90.0 - 122. - 200. - 305. - 187.7 - 186. - 286. - 134. - 165.0 - 253.0 - 77.2 - 88. - 169.3 - 471.7 - 238.3 - 268. - 493. - 319.1 - 49.8 - 367. - 273.2 - 241.0 - 207.3 - 207.5 - 120.5 - 275.5 - 94.76 - 59.3 (+4.1) - 113. - 96.9 - 113. - 306.9
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Standard Entropies of Hydration of Ions
No.
105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159
Ion
Co2+ c o 3 +
Co(NH3);+ Co(CN)i- Fe2+ Fe3+ Fe(CN)i - Fe(CN):- Pd2+ PdC1;- Pd( NH, ): +
RhC1;- Pt2+
Pt(NH3):+ IrCIi- IrCli- Mn2+ MnO; MnOi- TcO; Re- ReO, ReClg- Cr2+ Cr3 +
CrO:- Cr20:- MOO:- wo: V2 +
v3 +
vo2 +
vo: vo, vo: - HVOi- Zr4 +
HP+ sc3 +
Y3- Lu3+ Yb2+ Yb3+ Tm3+ Er3+
Dy3+ Tb3' Gd3+ Eu2+ Eu3 +
Sm2 +
Sm3 +
Pm3 +
ptc1;-
H O ~ +
*
*
* *
* *
*
* *
*
*
*
so (g) 179.52 179.30 435.2 464.8 180.32 173.99 482.5 469.8 185.42 412.2 4 10.0 410.4 194.6 425.7 415.8 421.9 4 16.7 173.78 277.8 291.1 288.5 183.30 294.1 426.8 181.74 178.96 28 1.4 379.7 291.1 196.6 182.09 177.84 225.9 259.0 266.9 284.8 296.1 165.23 173.63 156.37 164.90 173.38 173.24 190.53 194.26 195.87 196.19 195.50 193.50 189.33 188.90 180.48 183.12 186.75 189.46
-0 si con" (ad - 113. - 305.
146.0 232.6
- 137.7 - 315.9
270.3 95.0
- 184. 272. 307. 209. - 79. 218.7 44.8
222. 180. - 73.6
191.2 58.6
197.5 230. 201.3 61.
- 82. - 269.
50.2 261.9 27.2 40.6
- 74. - 307. - 133.9 - 42.3
50.
17. - 172.
- 509.3 - 465.7 - 255. -251. - 264. - 47. - 238. - 243. - 244.3 - 226.8 -231.0 - 226. - 205.9 - 8.
- 222. - 26. - 21 1.7 - 210.
Ahydr S P o n v
- 293. - 484. - 289.2 - 232.2 -318.0 - 489.9 - 212.2 - 374.8
(- 369.) - 140. - 103. -201. - 274. - 207.0
( - 37 1 .O) - 200. - 237. - 247.4 - 86.6 - 232.5 -91.0
(+ 47.)
- 366. - 264. - 448. -231.2 - 117.8 - 263.9 - 256.0 - 256. - 485. - 359.8 -301.3 - 217. - 457. - 279. - 674.5 - 630.3 -411. -416. - 437. - 220. - 429. - 437. - 440.2 - 423.0 - 426.5 - 420. - 395.2 - 197. - 402. - 209. - 398.5 - 399.
- 92.7
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No. 160 161 162 163 164 165 166 167 168 169 170 171 172 173 I74 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 I93 I94 195
Ion Nd3+
Ce3 +
Ce4 +
La3 +
Bk3+ Bk4+ Cm3 +
Am3+ AmO: AmOi- PU'+ Pu4 +
Puo; puo; +
Np3 ' Np4+ NPO: NpO:+ u3 +
u4 +
uo: uo:+
or3 +
Th4+ Ac3+ Be2+ Mg2+ Ca2 +
Sr2+ Ba2+ Ra2+ Li +
Na+ K + Rb+ cs+
* * * * * *
* *
* *
* *
*
SP (g)
190.10 188.93 185.49 170.60 170.49 199 .O 195.0 194.8 177.4 278. 274. 192.18 190.66 275. 270. 195.59 188.71 272. 266. 196.48 186.36 269. 260.4 176.91 176.63 136.26 148.68 154.93 164.72 170.35 176.58 132.99 147.98 154.63 164.4 1 169.86
Y. Marcus and A . Loewenschuss
$%l" ( a d
- 206.7 - 209. - 205. -301. -217.6 - 187.1 - 395. - 188.4 - 203.9 - 20.9 - 88. - 186.1 - 389. - 20.9 - 87.9 - 181.0 - 389. - 20.9 - 92.0 - 176.5 - 414. - 29. - 98.3 - 422.6 - 184.4 - 129.7 - 138.1 -53.1 - 32.6
9.6 54. 13.4 59.0
102.5 121.50 133.05
- 396.8 - 398. - 390. - 472. - 388.1 - 386.1 - 590. - 383.2 -381.3 - 299. - 362. - 378.3 - 580. - 296. - 358. - 376.6 - 578. - 293. - 358. - 373.0 - 600. - 298. - 358.7 - 599.5 -361.0 - 266.0 - 286.8 - 208.0 - 197.3 - 160.75 - 123. - 119.6 - 89.0 - 52.1 - 42.91 - 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-', Mi is the relative atomic mass of the ion, h is Planck's constant, NA is Avogadro's number, T i s 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.
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Standard Entropies of Hydration of Ions 87
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 o f j is obtained from the NBS Tables of Atomic Energy Levels.8 This is also the source for the energies of the higher levels, &k in cm-', from which the quantities u k = 1.43879&k/(T/K) and qk = (2jk + l)e-"k (both dimensionless) are obtained, and hence9
S:lec = R[ln c q k + c u k q k / c q k l (4)
The summation for T = 298.15 K extends over all those levels k that have &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 & 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 M a r ~ u s . ~ 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.
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88 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 = 0.24254[I/(amunm2)]-'(T/K)-' and I is the moment of inertia. The vibrational contribution for non-linear ions is
3 N - 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. Discussion of 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 $&,,(aq). For monoatomic ions these expressions include the following.
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Standard Entropies of Hydration of Ions
According to Powell and Latimer"
89
-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
(1 1) -0 S,,,,,(aq) = 197 - 6.441~~1 - [(r,!/nm) + J K-'rnol-l
where A = 0.13 nm for cations and A = 0.04 nm 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.
proposed. These include that by Connick and P ~ w e l l ' ~ For polyatomic ions that are oxyanions several other expressions were
(14) -0 S,,,,,(aq) = 182 - 1952, + 54n,JK-'mol-'
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. l 5 R. E. Connick and R. E. Powell, J . Chem. Phys., 1953, 21, 2206. l6 J . W . Cobble, J . Chem. Phys., 1953, 21, 1443.
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90 Y. Marcus and A . Loewenschuss
0.96 for linear triatomic ones, 1.34 for diatomic ones, and 0.77 for anions of ‘complex shape’. The latter are ‘dimeric’ anions, such as N,O:-, Cr,O:-, and C20i-. Another expression was proposed by Couture and Laidler:”
Szonv(aq) = 1.5R In M,, + 168 - 231~~1
-45.5zfn&’ - [(r,-&m) + 0.140]-1 JK-lmo1-l (16)
where nb is the number of ‘charge-bearing ligands’, 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 ‘complexes’ Cobble18 proposed the expression
Sz0,,(aq) = 205 - 4l.41zi(J;’(r,-,/nm)-’ + n,So(H,O, l)JK-‘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) = 69.91 J K-’ mol-’.
Finally, for oxycations the limited number of cases available still permits the correlation”
Si:on,(aq) = 1.5R In M, - 20 - 13.3zi(ri-,/nm)-’JK-’mol-’ (18)
based on the data for the vanadium, uranium, and transuranium element oxycations.
The expressions (10) to (1 8) are based on liner correlations of ,$’Eon, (aq) or of this quantity - 1.5R In Mri or - 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 & 5 to 15 JK-’mol-’, 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-’mol-’, so that none proved superior to the others. For ions where $&,,(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, & 6 J K-’ mol-’, 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. ’* J . W. Cobble, J . Chem. Phys., 1953, 21, 1446. l 9 Y. Marcus and A. Loewenschuss, unpublished results, 1984.
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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) = - 86 J 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 $,yonY(aq) of the latter, the least negative of the values given above for 0;- is preferred and entered in Table 1. No. 4 , H + and No. 6, H30+. Hydrogen ions, hydrated to various extents to H(H2O);, have distinct existences in the gas phase (see Section 4) and values of Sio(g) can be assigned to them. The nominal assignment of the conven- tional value of zero to SiOconv(aq) of the aqueous hydrogen ion does not permit a distinction between H+(aq) and H,O+(aq) (or any other of the hydrated hydrogen ion species). When the Sio(g) values for H+ and H,O+ are used,
2o R. M. Noyes, J . Am. Chem. SOC., 1964, 86, 971. 2' J. Jortner and R. M . Noyes, J . Phys. Chem., 1966, 70. 770. 22 B. E. Conway, J . S o h . Chem., 1978, 7 , 721. 23 JANAF, J . Phys. Chem. Re$ Data, 1982, 11, 695.
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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 K r e ~ t o v , ~ ~ " S,:o,v(aq) = 126 J 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 R y a b ~ k h i n . ~ ~ ~ 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), (1 5 ) 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$&,v is 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.
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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 $&(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 A s 4 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 f 2 J K - ' m o l - ' 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 S h o r ~ k h o v a ~ ~ of S&,(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.
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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
No. 62, SbOi-. The orthoantimonate anion is aquated in aqueous solutions to HiSbOip', therefore no value of ~ ~ o n v ( a q ) 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 G l a ~ i n a ~ ~ 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 0.3 172 V, after application of the Debye-Huckel correction to zero ionic strength of the measured results. The temperature coefficient dEo/dT = (1.30 From this the standard entropy change for this reaction, - 12.4 J 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 G l a ~ i n a , ~ ~ 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 I k ~ n n i k o v , ~ ~ 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 J IC '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 G l a ~ i n a , ~ ~ 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^.^
0.6 J K- ' mol-'.
0.15) 1 OP4 V K-' was also determined in that
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.
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Standard Entropies of Hydration of Ions 95
A report by L e ~ i e n , ~ for tetramethylammonium, s&(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 f 3, * 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.
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at 298.15 K. The contributions are Si& = 170.5, Si:ot = 33.6, and &",, = 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 Si&(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), ( l l ) , 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.
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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), 1 1 1.3 J 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 R y a b ~ k h i n , ~ ~ ~ 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- l a t i o n ~ , ~ ' . ~ ~ - 100 J K - ' 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 - 314 J 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.
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98 Y. Marcus and A . Loewenschuss
compilations. However, Vasil'ev and L ~ t k i n ~ ~ 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 3.4 J 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+. N o 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+. N o 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'/U1v couple was presented by S o b k o l o ~ s k i . ~ ~ 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,
5o J. Sobkolowski, J . Inorg. Nucl. Chem., 1962, 23, 81. Radiokhimiya, 1977, 19, 373.
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Standard Entropies of Hydration of Ions 99
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 s&(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 Si&, is 2.5R In (T/K). The translational and rotational contributions together are then
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 t ab~ la t ed .~ 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-
S,zans+rot(g, T) = S17rans+rot(g, 298.15K) + 33.26 In (T/298.15)JK-'mol-' (19)
" 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.
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100 Y. Marcus and A . Loewenschuss
Table 2 Parameters for equation (20): aj(T)/J K-I mol-' = ajo + ajl T + aj2T2 and bj ( T ) = bjo + bj, T + b,, T2, valid to 423 K (with possibly larger errors, to 573K). j 4 0 aj I 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 i t 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 ( a q , 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':
where a, and bJ are temperature-dependent parameters that, fortunately, are not ion-specific, but common to groups of ions j. The values of aj and bJ were 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
$yix(aq, T ) = a j ( ~ ) + bJ(T)Sl%S(aq> Tr) (20)
it follows that (22) -0
Siconv(aq, T ) = S:onv(aq, Tr) + t(T/Tr) - 11c;jconv(aq,
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.
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Standard Entropies of Hydration of Ions 101
found, e.g., in the work of Helgeson et al.” for some 45 ions. Khodakovskiiss proceeded, however, to give an empirical expression for the heat capacity [on the convention that Ciconv(H+, aq) = 01:
(23) -0 Cplconv(aq, Tr> = a; - ~ I Z , I - +S%w(aq, ~ r )
Here, again, as and a’, 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 a’, = 124.7, and for all anions d, = 31 1.3, all values in JK-lmol-’. 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 T’ 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 e q ~ a t i o n , ’ ~ 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
Ahydr So( T ) = ( N A e 2 / 8 ? t ~ 0 R,)E( T ) - 2 ( d ~ / d T ) p - C, (26)
No actual application of this set of equations for the calculation of entropies of hydration was presented by the
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)
however.
-0 Stconv(aq , = sEonv(aq, ~ r > + c:conv(aq, Tr) ln (T/Tr)
+ C, In [(T - o , ) / (~r - o,) l+ ( N A ~ ~ z , ~ / ~ ~ & 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
57 J . W. Cobble and R. C. Murray, jun., Furaduy Disc. Chem. Soc., 1978, 64, 144. referred to is on pp. 1414, 1434, and 1435).
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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 (rc is 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 Ci and Oi were t a b ~ l a t e d , ~ ~ as were those of the coefficient of the difference of the Y’s in the Born term and the values of c&,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 ~ / d T ) ~ ] .
The application of high pressure mass spectrometry has made possible measurements on the extent of the reaction
4 Entropies of Hydration of Ions in the Gas Phase
X(H,O):,- I (g) + H*O(g) + X(H,O):,(g)
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 H 3 0 + . 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.58 Improved 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-
(28)
’* 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.
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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 A1,2S0 and 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 A1,2S0 was 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- , ,nSo beyond, say, n = 4, due to the larger freedom of motion of the outer molecules. An increase of the absolute value of A4,5So would 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 A4,5 So for 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 M e ~ t - N e r , ~ ~ ~ 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.
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.
69 (a) J. D. Payzant, A., J. Cunningham, and P. Kebarle, Can. J . Chem., 1973, 51, 3242; (b) M. Meot-Ner,
70 P. M. Holland and A. W. Castleman, jun., J. Chem. Phys., 1982, 76, 4195.
Zaved., Khim. Khim. Tekhnol., 1975, 18, 554.
I. N. Tang and A. W. Castleman, jun., J. Chem. fhys. , 1972, 57, 3638.
J . Am. Chem. Soc., 1984, 106, 1265.
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104 Y. Marcus and A . Loewenschuss
stages of the former ion could not be attained. It was also argued that for both of these ions a co-ordination number of four was favoured.
The number of negative ions for which the extent of hydration in the gas phase has been studied is somewhat larger. The thermodynamic functions for the halide ion-water clusters were determined, first by Arshadi et aL7' and then by Keese et aZ.72 for the chloride and iodide-water clusters. The two sets of values agree well, except for the first hydration stages, but the discrep- ancies were ascribed to differences in the mode of determination of the bond energies rather than in the Gibbs free energies.72 The thermodynamic func- tions were also determined for several of the hydration stages of OH-, O F , CN-, NO,, NO;, and HCO, .73-75 The relevant entropy data are presented in Table 3.
Although in some of these investigations the probable error in the entropy value reported was estimated, a better feeling for the accuracies attained can be obtained from the spread of the values shown in Table 3 for a given ion-water cluster.
Mruzik et aZ.76 attempted to determine theoretically the thermodynamic variables of ion-water clusters by Monte-Carlo methods based on 'second generation' potential functions, which adjusted the model parameters to fit ab-initio Hartree-Fock computations of ~ a t e r - w a t e r ~ ~ and i ~ n - w a t e r ~ ~ interactions. The entropy values for reaction (28) for the ions Li+, Na', K+, F-, and C1- are listed in Table 3, where they are compared with the experi- mental values. It may be noted that both experimental and computed values are rather weak functions of the cluster size and the type of ion. Nevertheless, the initial trend of the entropy becoming more negative with increasing cluster size is considered to be indicative of the loss of spatial freedom due to crowding. Both experimental and computed values approach, for larger n values, the entropy of a surface water molecule in bulk water (- 108.8 J K-' mol-').79 An evaluation of A0,] So at 298.15 K by statistical methods on the basis of molecular data was carried out by Ushanova et aZ.*' for Li+, Na', and K+, and the calculated values are also shown in Table 3. A similar calculation for clusters of the alkali metal ions and the halide ions with up to 6 water molecules, based upon the normal modes derived from an electrostatic model was reported by Bekmuratova et aZ.,81 and the results are shown in Table 3.
" M. Arshadi, R. Yamdagni, and P. Kebarle, J . Phys. Chem., 1970, 74, 1475. 72 R . G. Keese and A. W. Castleman, jun., Chem. Phys. Lett., 1980, 74, 139. 73 J. D. Payzant, R. Yamdagni, and P. Kebarle, Can. J . Chem., 1971, 49, 3308. 74 M. Arshadi and P. Kebarle, J . Phys. Chem., 1970, 74, 1483. 75 R. G . Keese, N. L. Lee, and A. W. Catleman, jun., J . Am. Chem. Soc., 1979, 101, 2599. 76 M. R . Mruzik, F. F. Abraham, D. E. Schreiber, and G. M. Pound, J . Chem. Phys., 1976, 64, 481. 77 G. C. Lie and E. Clementi, J. Chem. Phys., 1975,62, 2195. 78 H. Kistenmacher, H. Popkie, and E. Clementi, J . Chem. Phys., 1973, 59, 5842. 79 P. J . Good, J . Phys. Chem., 1957, 61, 810.
N. I . Ushanova, I. N. Godnev, R. M. Kotomina, and A. M. Aleksandovskaya, Zh. Fiz. Khim., 1981,55, 1864. E. M. Bekmuratova, S. L. Pozharov, and P. K. Khabibullaev, Zh. Fiz. Khim., 1983, 57, 1798 and 1928. 8 1
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Standard Entropies of Hydration of Ions 105
Table 3 Standard molar entropies of ion-water cluster formation at 298.15 K a. Proton-water clusters, H(H20)T-l + H 2 0 = H(H20),
n = 1 2 3 4 5 6 7 8 Ref. - An- ,$O/J K - ' mol-'
139.4 121.4 118.4 136.4 126.8 123.9 113.0 58 101.7 91.7 114.3
129.8 111.7 132.7 -4.2 58.6 117.2 117.2 71.1 72.0 70.6 83.7 68.2
140.6 82.9 85.0 101.7 90.8 118.9 97.9 104.6 109.2
104.2 124.1 121.4 118.1
b. Water clusters of other ions ion n = 1 2 3 4
- An-,,So/J K-lmol-' Li +
Na'
K +
Rb'
cs+ NH:
c u +
F- Ag+
c1-
Br-
I -
OH- 0; CN- NO; NO, HCO;
96.3 94.6 94.6 90.0
86.6 90.8
102.5 90.4 92.9 79.5 85.8 92.1 92.1 88.7 87.9 82.4 96.7
118.9 72.8 90.4
100.4 69.1 82.4 68.6 89.6 77.0 89.1 68.2 80.8 85.0 87.0 84.1 82.9 87.9 79.9
100.9
88.3 106.3
92.9
99.6
130.2 101.3 110.9 87.5
92.9 108.8 92.9
100.4 91.7
93.3 78.3
100.2 123.9 87.0 85.8 76.2
104.6 95.8
103.8 79.5 85.0 95.4 88.7
105.0
99.2
121.8
104.2 112.2
91.7
99.2
121.4 96.3
100.9 89.6
100.4 99.2 99.2 92.1
105.0 88.7
100.0 90.4 85.4
100.2 116.8 97.1 93.7 80.4 98.8
103.8 93.3 89.1 87.9 86.6
136.2*
88.7
126.4
125.1 111.3
104.6
106.3
126.0 103.4 115.5 87.5
103.8 107.6 106.3 102.5 114.3 86.2
126.4 123.5 154.4 110.1 121.4 108.0 103.8 89.1
104.6 112.2 104.2
98.8
139.4
5
131.4 121.4
117.6 110.9 105.9
136.0 105.5 106.7 85.4
107.6 111.3
105.0 93.7
113.0 121.8 126.8 128.5 105.8 130.2
81.6 107.1
107.6
100.9
6
133.9 133.9
108.8
97.9
145.2 107.6 130.2 89.1
128.9
123.9
134.8
88.3 141.5
99.2 125.6
123.9
119.7
59 60 61" 61b 62 63 64 65
Ref.
66 76' 80' 66 68 76' 80' 81' 66, 67 81' 76' 80' 66 81' 66 81' 69a 69b 70 70 71 76' 81' 71 72 76' 8 1' 71 8 1' 71 71 81' 73 74 73 73 73 75
'Methane buffer gas; 'propane buffer gas; 'theoretical calculation; +recalculated to correct error pointed out by F. C. Fehsenfeld and E. E. Ferguson, J . Chern. Phys., 1974, 61, 3181
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106 Y. Marcus and A . Loewenschuss
5 Absolute Entropies of Hydration The conventional entropies of hydration are based, according to equations (2) and (3) , on the convention that Sioconv(aq) = 0 for H+(aq) at all tem- peratures. An assignment of an absolute value to this ion at some tem- perature permits, therefore, the calculation of absolute entropies of hydra- tion for all ions at this particular temperature. Most of the effort in this direction was limited to the one temperature, T = 298.15 K.
Within the framework of an evaluation of various properties of individual ions in solution, Conway22 dealt also with the standard partial molar entropy of the aqueous hydrogen ion. He reviewed most of the literature available at the time, stressing the results from mainly two methods.
The method involving mercury at the potential of zero charge (pzc) was employed by Lee and Tai,82 who measured the difference in the e.m.f. of the cells
Hz (g), Pt/HCKaq)/Hg2 c12 (s)/Hg(l)
at various temperatures. Extrapolation of the e.m.f. difference to the stan- dard state conditions and taking the temperature coefficient yielded the standard molar entropy for the reaction 4H2(g) + H+(aq) + e-(Hg), hence the value of sEbs(H+, aq).
(30)
E a ~ t m a n ~ ~ measured the e.m.f. of the thermocell
Ag(s)/AgBr(s)/KBr(aq)/AgBr(s)/Ag(s) (31)
T, T2
Ideally, this should yield the entropy of the reaction AgBr(s) + e- -+ Ag(s) + Br-(aq) and the value of sEbbs(Br-, aq). In practice, corrections for the entropy of transfer of ions across the temperature gradient in the solution and of electrons over that in the external circuit must be applied.
Several other authors also addressed themselves to this problem, some of them using thermocells with various methods of making the required correc- tions, others employing different methods altogether. The resulting values of Zbs (H+, as), obtained directly from the measurements or indirectly from the value of another ion via the additivity rule, are shown in Table 4.
So called ‘correspondence plots’ were employed by some of the authors. Gurney84 found a linear correlation of the partial molar entropy of aqueous ions with their Jones-Dole B-coefficients of the viscosity (partitioned into ionic values on the basis of BK+ = Bcl-). The values for cations and anions are located on a single straight line only if sEbs(H+, aq) is assigned the value shown in Table 4. Criss and Cobble8’ made a similar evaluation on the basis of a correlation with partial molar heat capacities.
a’- F. H. Lee and Y. K. Tai, J. Chin. Chem. SOC., 1941, 8, 60. 83 M. Eastman, J. Am. Chem. SOC., 1928, 50, 283 and 292. 84 R. W. Gurney, ‘Ionic Processes in Solution’, McGraw-Hill, New York, 1953, Ch. 10. 85 C. M. Criss and J. W. Cobble, J. Am. Chem. SOC., 1964, 86, 5390.
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Standard Entropies of Hydration of Ions 107
Table 4 Estimates of the absolute partial molar entropy of the aqueous hydro- gen ion at 298.15 K
Author Method SEbs(H+, aq)/JK-'mol-l Eastmang3 Lange and Hesse" Latimer et Lee and TaiB2 Cockroft and Halliwb Goodrich et al.89 Gurneyg4 deBethune et al.'w" NoyesB6 Criss and Cobble" Lin and Breck" Breck et 01.~' Breck and Lin92 Ikeda93 DeLigny et al.94 Nedermeyer et al.95 Jalenti et al.96 Payton et Tremain et al.98 Jenkins and P r i ~ h e t t ~ ~
thermocells thermocells extrapolation vs. I / r Hg electrode at pzc thermocells thermocells correlation with B thermocells extrapolation vs. 1 / r correlation with C, isentropic redox couples thermocells thermocells ionic mobilities fic/foc couple fic/foc couple correlation with H thermocells thermocells competition principle
- 20.9* - 19.6* - 6.3 - 22.6* -26.3* at 303K
-8 .8 f 1.7 - 23.0* - 18.7 - 13.8 - 20.0* - 23.8 - 18.8 to -24.3 - 23.0* - 21.3 or - 22.6* -20.1 & 5.0 -20 f 5 - 20.9 - 19.7 f 0.4 - 19.4 2.5 -2.6 to - 8.1
* Values used by Conway22 to obtain the mean value - 22.2 f 1.4.
Noyesg6 employed extrapolations of - zi (sEonv(aq) - AS',J against reciprocal ionic radii of the cations K+, Rb+, Cs+, and the halide anions to zero reciprocal radius. The term subtracted, AS:,, = - 26.7 J K-' mol-', represents the entropy change for the change in the standard state volumes on hydration. The common intercept for the extrapolation of the values for the cations and anions according to second degree curves in (l/ri) is AS; = 79.0 J K - ' mol-'. This is the standard entropy change for the hypothetical reaction +H,(g) - H+(aq) + e-(g), and from it the value quoted in Table 4 was derived. An alternative set of values of ASZut, based on an interpolation in the entropies of solution of the inert gases at the values of ri of the ions, led to no common intercept for the cations and the anions. The older estimate of sOabs(H+, as) by Latimer et a1.87 is similar in principle, but they used 'effective radii' for the ions. These were obtained by the addition of 0.01 nm to the crystal ionic radii of the halide anions and of 0.085nm to those of the alkali metal cations, the same values that were used for fitting ionic Gibbs free energies of hydration. A straight line was obtained for both cations and anions, when plotted against the reciprocals of these effective radii, when sOabs(H+, aq) was assigned the value shown in Table 4.
86 W. M. Noyes, J. Am. Chem. Soc., 1962, 84, 513. '' W. M. Latimer, K. S. Pitzer, and A. Slansky, J . Chem. fhys. , 1939, 8, 107. 88 E. Lange and T. Hesse, Z . Elektrochem., 1933, 39, 374. 89 J. C. Goodrich, F. M. Goyan, E. E. Morne, R. E. Preston, and M. B. Young, J . Am. Chem. SOC., 1950,
72, 441 I . W. Breck, G. Cadenhead, and M. Hammerli, Trans. Furaday Sor., 1965, 61, 37.
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108 Y . Marcus and A . Loewenschuss
Following earlier work of Lin and Breck" on isentropic redox couples, DeLigny and c o - w ~ r k e r s ~ ~ , ~ ~ considered that the redox potential of the ferric- inium ion/ferrocene couple ('fic/foc') should be related to the difference in the Gibbs free energy of hydration of these species. This, in turn, depends only on the electrostatic contribution to the Gibbs free energy of hydration, the neutral component of this quantity being the same. The temperature coef- ficient of this potential, in the range from 288 to 318 K, was used for the evaluation of single ion standard partial molar entropies, and also of the temperature coefficient of the surface potential of water against vacuum: dXH20/d T = - 0.45 f 0.15 mV K-'. The value of the surface potential itself was later revised by the authors (from - 0.3 f 0.1 V94 to 0.04 0.1 V9'), but the temperature coefficient remained the same as did the value of
The tempeature coefficient of the surface potential is a component of the E b s ( H + , as).
'real' partial molar entropy of an ion:
S:daq) = S%,(aq> - ~ , F ( % , o / W , (32)
where F is the Faraday constant. The temperature coefficient of the e.m.f. of the cell
Ag(s)/AgCl(s)/KCl(aq jet)/air/KCl(aq)/AgCl(s)/Ag(s) (33)
measured by Randles and Schiffrin'" was used by Schriffrin'02 for the deter- mination of the 'real' standard partial molar entropy of the aqueous chloride ion:
S:eal(Clp, aq) = lim [So(AgC1, s) - So(Ag, s) + F(dE/dT) conc-0
+ R In a,(KCl, aq) - tt(KC1, aq)/T] (34)
where a+ is the mean ionic activity and t is the relative partial molar heat content.-The resulting value was $:eal(Cl-, aq) = 36 4 2 J K-' mol-I, from which Si:eal(H+, as) = 19 f 2 J K ' m o l - ' and ( a ~ ~ , ~ / a T ) , = -0.39 k 0.04mVK-' were derived, the latter value being not far from the value obtained by DeLigny et aZ.94
The review by Conway22 considered the results from the thermocells to have 'excellent' reliability, in the senses that the method was theoretically well
91 J. Lin and W. Breck, Can. J . Chem., 1965, 43, 766. 92 W. Breck and J. Lin, Trans. Faraday Soc., 1965, 61, 2223. 93 T. Ikeda, J . Chem. Phys., 1965, 43, 3412. 94 C. L. DeLigny, M. Alfenaar, and N. G. van der Veen, Recl. Trav. Chim., 1968, 87, 585. 95 H. J. M. Nedermeyer and C. L. DeLigny, J . Electroanal. Chem. Interfacial Electrochem., 1974, 57, 265. 96 R. Jalenti and R. Caramazza, J . Chem. SOC., Faraday Trans. I , 1976,72, 215. 97 A. D. Payton, E. J . Amis, M. S. Showell, R. W. Smith, and B. D. Koplitz, J . Electrochem. SOC., 1980,
98 P. R. Tremain, M. H. Sargent, and G. J . Wallace, J . Phys. Chem., 1981, 85, 1977.
loo (a) A. J . deBethune, T. S. Licht, and N. Swendeman, J. Electrochem. SOC., 1959, 106, 616; (b)
lo' J. E. B. Randles and D. J. Schiffrin, J . Electrochem. SOC., 1965, 10, 480. '02 D. J. Schiffrin, Trans. Faraday Soc., 1970, 66, 2464.
127, 2157.
H. D. B. Jenkins and M. S. F. Pritchett, J . Chem. SOC., Faraday Trans. 1, 1984, 80, 721.
H. D. Cockcroft and J. R. Hall, J . Phys. Chem., 1950, 54, 731.
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Standard Entropies of Hydration of Ions 109
founded, that the errors were small, and that several applications were in good mutual agreement. This judgment was made in spite of the fact that the method required an extrathermodynamic estimate of the division of the entropy of transport into the ionic contributions. Lower marks were given to the methods involving the potential of zero charge and the correspondence plots, and still lower ones to methods involving extrapolation with respect to l / r , . The data from the more reliable methods were averaged to yield the recommended value S,O,bs(H+, aq) = - 22.2 1.4 J K-' mol-' at 298.15 K, see Table 4.
It ought to be pointed out that more recent determinations with thermo- cells yielded the somewhat less negative value of - 19.7 +_ 0.4 and - 19.4 2.5 J K-' mo1-' for this quantity.The latter overlaps within its wide margin of error the value recommended by Conway,22 the former is marginally outside it, due to its narrow assigned uncertainty. The correction steps for the transported entropy by the individual ions and the assumptions required by them, leading to these newer values, are not compelling, however. For the present, until substantially more reliable means for the division of the trans- ported entropy between the ions are devised, the estimate arrived at by Conway22 should stand.
Once Szbs(Ht, aq) has been fixed at 298.15 K, application of the additivity principle permits the calculation of the absolute standard partial molar entropies of all the ions for which conventional values are known at this temperature:
Si$:,,(aq) = Si%w(aq) + z,Si%:bs(H+, as) = Si%v(aq) - 22-22, (35)
The absolute standard molar entropies of hydration at this temperature are obtained for all these ions, see Table 5 , from:
'hydrSi;bs = 'hydrS:conv - 22.22~ (36)
Little is definetly known concerning the absolute standard molar partial entropy of the hydrogen ion at temperatures other than 298.15 K. Criss and Cobble53 found that their correspondence plots yielded consistent values for cations and anions at the temperatures 333.15, 373.15, and 423.15K, if SOabs(H+, as) were assigned certain values at these temperatures. (The negatives of the values can be obtained from the quadratic equation in the footnote of Table 2). These have not been confirmed by direct measurements on thermocells. The work of Tremain et aZ.98 dealt with the absolute heat capacity change obtained from thermocell measurements over the temperture range of 298.15 to 363.15 K, but no direct calculations of the partial molar entropy were made.
The isotope effect on the transfer of (hydro/deutero)chloric acid from light water to heavy water was measured by A b r o z i m ~ v . ' ~ ~ The isotope effect on the transfer of the chloride anion between these two kinds of water was estimated from the measured isotope effect on the transfer of sodium chloride, on the assumption that the structural effects of Na+ ions on '03 V. K. Abrozimov, Radiokhimiya, 1972, 14, 916.
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110 Y. Marcus and A . Loewenschuss
Table 5 Absolute standard molar entropies (J K-' mol- ' 1 of hydration of ions, their radii" and the Born and other contributions to the entropy of hydration
No. , ion abs hydr.
Monoatomic cations: 4 H+
191 Li+ 192 Na+ 193 K + 194 Rb+ 195 Cs+ 96 Cu+ 98 Ag+ 90 Tl+
185 Be2+ 186 Mg2+ 187 Ca2+ 188 sP+ 189 Ba2+ 190 Ra2+ 157 Sm2+ 155 Eu2+ 147 Yb2+ 135 V2+ 129 Cr2+ 122 Mn2+ 109 Fe2+ 105 Co2+ 104 Ni2+ 113 Pd2+ 117 Pt2+ 97 cu2+ 92 Zn2+ 93 Cd2+ 94 Hg2+ 77 Sn2+ 80 Pb2+ 85 ~ 1 3 +
144 Sc3+ 145 Y3+ 164 La3+ 162 Ce3+ 161 Pr3+ 160 Nd3+ 159 Pm3+ 158 Sm3+ 156 Eu3+ 154 Gd3+ 153 Tb3+ 152 Dy3+ 151 Ho3+ 150 Er3+ 149 Tm3+ 148 Yb3+
- 131 - 142 - 111 - 74 - 65 - 59 - 143 - 117 - 72 - 310 - 331 - 252 - 242 - 205 - 167 - 253 - 241 - 264 - 300 - 308 - 292 - 362 - 337 - 351
(-413) - 318 - 320 - 318 - 285 - 252 - 230 - 209 - 538 - 478 - 483 - 455 - 457 - 465 - 463 - 466 - 465 - 469 - 462 - 487 - 493 - 490 - 507 - 504 - 496
r/nm
0.000 0.059' 0.102 0.138 0.149 0.170 0.096' 0.1 15 0.150 0.027' 0.072 0.100 0.113 0.136 0.143' 0.1 19d 0.1 17 0. 10Sd 0.079 0.082 0.0830 0.0780 0.0745 0.069 0.086 0.080' 0.073 0.075 0.095 0.102 0.093b 0.118 0.053 0.0745 0.090 0.1045 0.101 0.0997 0.0983 0.097 0.0958 0.0947 0.0938 0.0923 0.0912 0.090 1 0.0890 0.0880 0.0868
ASi",o, hydr.
- 14.5 - 12.0 - 10.6 - 9.7 - 9.5 - 9.0 - 10.8 - 10.3 - 9.4 - 52.9 - 46.1 - 42.7 -41.3 - 39.0 - 38.4 -41. - 40.9 - 42. - 45.2 - 44.9 - 44.7 - 45.4 - 45.8 - 46.5 - 44.4 - 45.1 - 46.0 - 45.7 - 43.3 - 42.5 - 43.5 - 40.8 - 109.7 - 103.1 - 98.8 - 95.0 - 95.9 - 96.2 - 96.6 - 96.9 - 97.2 - 97.5 - 97.8 - 98.1 - 98.4 - 98.7 - 99.0 - 99.3 - 99.6
AS:* hydr.
- 90 - 103 - 74 - 38 - 29 - 23 - 105 - 80 - 36
-231 - 258 - 183 - 174 - 140 - 102 - 185 - 173 - 195 - 228 - 236 - 220 - 290 - 265 - 278
(- 342) - 246 - 247 - 245 -215 - 183 - 160 - 142 - 402 - 348 - 358 - 333 - 334 - 342 - 340 - 342 - 341 - 345 - 337 - 362 - 368 - 364 - 381 - 378 - 370
4"rn s:
(- 121) - 74 - 57 - 57 - 51 - 87 - 75 - 59
(-481) - 230 - 167 - 153 - 129 - 125 - 148 - 151 - 169 - 216 - 208 - 206 - 220 - 231 - 250 - 204 - 223 - 236 - 230 - 185 - 175 - 189 - 151
(- 489) (- 348) - 292 - 254 - 263 - 267 - 271 - 275 - 278 - 282 - 285 - 290 - 293 - 297 - 301 - 304 - 309
4 t m c s,P
0 + 19 + 28 + 28 - 18 -5
+ 23
- 28 - 16 - 21 - 11 + 23 - 37 - 22 - 26 - 12 - 28 - 14 - 70 - 34 - 28
- 23 - 11 - 15 - 30 -8
+ 29 + 9
- 66 - 79 - 71 - 75 - 69 - 67 - 63 - 53 - 42 - 72 - 75 - 67 - 80 - 74 - 61
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Standard Entropies of Hydration of Ions
No . , ion ASiabs hydr.
146 Lu3+ 184 Ac3+ 179 U3+ 175 Np3+ 171 Pu3+ 168 Am3+ 167 Cm3+ 165 Bk3+ 136 V3+ 130 Cr3+ 110 Fe3+ 106 Co" 88 Ga3+ 89 In3+
63 Bi3+ 163 Ce4+ 142 Zr4+ 143 H€'+ 183 Th4+ 180 u4+ 176 Np4+ 172 Pu4+ 166 Bk4+ 78 Sn4+
91 ~ 1 3 +
Monoatomic anions 0 e-- 8 F-
10 c1- 15 Br- 20 1- 23a At-
126 Re- 1 02-
24 S2- 37 Se2-
Diatomic ions
195 Hg;+
25 S;- 5 OH-
34 SH- 40 SeH- 11 c10- 17 BrO- 46 NO+ 61a SbO+
137 V 0 2 + 69 CN-
2 0, 3 0;-
- 504 - 428 - 440 - 443 - 445 - 448 - 450 - 453 - 552 - 514 - 557 - 551 - 560 - 386 - 434 - 394 - 561 - 763 - 719 - 688 - 689 - 667 - 669 - 679 - 375
36 - 137 - 75 - 59 - 36 - 19 (69)
- 185 - 122 - 149
- 252 - 82 - 256 - 150 - 161 - 98 - 102 - 152 - 163 - 323 - 232 - 382 - 80
r/nm
0.0861 0.1 18' 0.104 0.102 0.101 0.100 0.098 0.096 0.064 0.0615 0.0645 0.061 0.062 0.079 0.088 0.102 0.080 0.072 0.07 1 0.100 0.097h 0.095' 0.093' 0.093 0.069
0.133 0.181' 0.1 96' 0.220' 0.228'
0.140 0.184' 0.1 98'
0.39' 0.158' 0.173' 0.28' 0.133' 0.207c 0.205' 0.21' 0.23' 0.14' 0.24' 0.22' 0.191'
A$Ln hydr.
- 99.8 - 91.8 - 95.2 - 95.7 - 95.9 - 96.2 - 96.7 - 97.2 - 106.2 - 107.0 - 106.1 - 107.2 - 106.8 - 101.8 - 99.3 - 95.7 - 180.4 - 184.5 - 185.1 - 170.9 - 172.3 - 173.2 - 174.2 - 174.2 - 186.1
- 9.8 - 8.8 - 8.5 -8.1 - 7.9d
- 38.7 - 35.0 - 34.0
- 24. - 9.3 - 35.8 - 29. - 9.8 - 8.3 - 8.4 - 8. - 8. - 10. - 8. - 8. - 8.6
ASP* hydr.
- 378 - 309 - 318 - 321 - 322 - 325 - 326 - 329 -419
(- 380) - 424
(-417) - 427
(- 257) - 308 - 272
(- 354) - 552
(- 507) - 491 - 490 - 467 - 468 - 478
(- 162)
- 101 - 39 - 24
- 1 16
- 120 - 61 - 88
- 201 - 46 - 194 - 95 - 124 - 63 - 67 - 117 - 128 - 286 - 175
(- 347) - 45
4 l l m s P
-311 - 228 - 259 - 264 - 267 - 270 - 275 - 281 - 409 - 426 - 406 -431 - 425 - 336 - 306 - 263 - 447 - 493 - 506 - 360 - 371 - 379 - 387 - 388 - 517
- 54 - 42 - 43 - 39 - 39
- 108 - 86 - 86
- 46 - 48 - 92 - 59 - 53 - 36 - 41 - 38 - 37 - 54 - 36 - 76 - 38
111
4 m l c s,P
- 67 - 101 - 59 - 57 - 55 - 55 - 51 - 48 - 10
- 18
-2
-2 -9
- 59
- 131 - 119 - 88 - 91 - 90
- 47 + 3
+ 19 + 38 + 55
- 12 + 25 -2
- 155 + 2
- 102 - 36 - 71 - 27 - 26 - 79 -91 - 232 - 139
-7
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No., ion
Triatomic ions: 45 N; 26 S:- 16 Br; 21 I;
9 HF, 7 HO,
81 BO; 86 A10; 47 NO: 48 NO; 59 AsO, 12 ClO,
138 VO: 181 UO: 177 NpO: 173 PuO: 169 AmO: 182 UO:+ 178 NpO:'
170 AmO:' 70 NCO- 71 NCS-
174 puo:+
99 AgCl,
Te t raa tom ic ions:
6 H30+ 13 ClO; 18 BrO;
49 NO;
64 C0:- 27 SO:- 38 SeOi- 43 Te0:- 50 N,O:- 66 HCO,
22 10,
139 VO;
abs hydr.
- 82 - 197 - 89 - 73 - 97 - 183 - 23.1 - 216 - 329 -91 - 205 - 134 - 324 - 320 -315 - 318 - 321 - 403 - 402 - 402 - 406 - 90 - 66 - 37
-215 - 80 - 95 - 148 - 76 - 195 - 245 - 249 - 227 - 237 - 185 - 102
Tetrahedral oxyanions 14 C10; - 57
23 10; - 53 123 MnO; - 64 125 TcOy - 69 127 ReO; -71
39 Se0:- - 183 44 TeOi- - 206
131 Cr0;- - 187
19 BrO; - 60
28 SO:- - 200
133 MOO:- - 220 134 WOf - 212 124 Mn0:- - 188
r/nm
0.195" 0.33' 0.40' 0.47' 0.172' 0.18' 0.24' 0.26' 0.18' 0.192' 0.29' 0.25' 0.26' 0.29' 0.29' 0.30' 0.30' 0.28' 0.28' 0.29' 0.29' 0.203' 0.21 3' 0.41'
0.13e
0.191g O.18lg 0.179' 0.182' 0.178' 0.20' 0.239" 0.25' 0.17' 0.169'
0.200g
0.240' 0.25' 0.24Y 0.2409 0.25' 0.26' 0.2309 0.243g 0.254g 0.2409 0.254g 0.27' 0.25'
ASPBOrn hydr.
- 8.5 - 27. - 6. - 5. - 9.0 - 9. - 8. - 8. - 9. - 8.6 - 7. - 8. - 8. - 7. - 7. - 7. - 7. - 29. - 29. - 29. - 29. - 8.4 - 8.2 - 6.
- 10. - 8.5 - 8.6 - 8.8 - 8.8 - 8.8 - 35.5 - 34. -31.3 -31 - 36 - 9.0
- 7.8 -8 - 7.7 - 8.0 -8 -8
-31.8 -31.1 - 30.4 - 29.0 - 30.4 - 30 -31
ASP* hydr.
- 47 - 143 - 56 - 42 - 61 - 147 - 196 - 181 - 293 - 56 - 172 - 99 - 289 - 286 - 281 - 284 - 287 - 347 - 346 - 346 - 350 - 55 - 31
4
- 178 - 44 - 60
-112 - 41 - 160 - 183 - 188 - 169 - 179 - 122 - 66
- 22 - 26 - 19 - 30 - 34 - 36 - 142 - 125 - 149 - 129 - 162 - 155 - 130
4 r i m SP
- 40 - 52 - 22 - 19 - 45 - 42 - 32 - 31 -44 - 41 - 29 - 33 - 32 - 31 - 31 - 30 - 30 - 64 - 64 - 62 - 62 - 38 - 38 - 21
- 54 - 42 - 45 - 49 - 46 - 47 - 94 - 85 - 73 - 71 - 98 - 47
- 36 - 35 - 35 - 36 - 35 - 34 - 75 - 72 - 70 - 72 - 69 - 66 - 69
4 t m c SP
-7 - 91 - 34 - 23 - 16 - 105 - 164 - 150 - 249 - 15 - 143 - 66 - 257 - 255 - 250 - 254 - 257 - 283 - 282 - 284 - 288 - 17 + 7 + 25
- 124 -2 - 15 - 73
+ 5 - 113 - 89 - 103 - 96 - 108 - 24 - 19
+ 14 + 9 + 16 + 6 + I - 2 - 67 - 53 - 79 - 57 - 95 - 89 - 61
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Standard Entropies of Hydration of Ions
No. , ion Asi abs hydr.
55 Po;- -421 60 As0:- - 385 62 Sb0:- - 387
140 V0;- - 378
Other oxyanions: 29 S20:- - 180 30 S,O:- - 183 31 S,O;- - 167 32 S,O;- - 53 33 s,o;- - 55
132 Cr,O$- - 73
56 P20:- - 371 65 C,O:- - 205
Protonated oxyanions:
35 HSO; - 105 41 HSeO; - 126 67 HCO; - 137 36 HSO; - 129 42 HSeO; - 124 57 HPOZ- - 272 61 HASO:- - 260
141 HVOi- - 235 58 H,PO; - 166 84 B(0H); - 147 87 Al(0H); - 246
Further polyatomic cations:
51 NH: - 112 52 N,H: - 102
54 NH,OH+ - 102 72 CH,NH: - 112
74(c2 H, 14 N +
100 Ag(NH,): (- 18)
53 N,H;+ - 168
73(CH,),N+ - 144 - 222
75 (C,H,),N+ - 327
107 Co(NH,)i+ - 356 I I5 Pd(NH,)i+ - 147 1 I9 Pt(NH,):+
Further polyatomic anions:
(- 4 15)
68 CH,CO; - 170 76 SiFi- - 143 79 SnF2- - 90
83 BF; - 66 101 Ag(CN), -91
103 Au(CN; -91 108 Co(CN)i- - 166 111 Fe(CN)i- - 146 112 Fe(CN);f- - 286
82 BH; - 55
102 AuCI; - 75
rlnm
0.238g 0.248R 0.2609 0.260"
0.25' 0.25' 0.25' 0.29' 0.31' 0.30' 0.21' 0.32'
0.17' 0.21' 0.156' 0.19' 0.21' 0.20' 0.21' 0.22' 0.20' 0.23' 0.29'
0. 148c 0.19' 0.19' 0.19' 0.20' 0.286 0.33P 0.372' 0.30' 0.42' 0.29' 0.31'
0.162' 0.259' 0.29' 0.193' 0.232" 0.34' 0.33" 0.32' 0.43' 0.44' 0.45'
A G B O r n hydr. - 70.5 - 69.2 - 68 - 67.7
- 31 - 31 - 31 - 29 - 28 - 112 - 33 - 27
-9 -8 - 9.3 -9 -8 - 34 - 33 - 33 -9 -8 -7
- 9.5 -9 - 35 -9 -9 - 7.2 - 6.6 - 6.2 -7 - 52 - 29 - 28
- 9.2 - 30.1 - 29 - 8.6 - 7.9 -7 -7 -7 - 51 - 51 - 89
AS:* hydr. - 324 - 289 - 290 - 284
- 122 - 125 - 109
3 0
- 232 - 145 - 20
- 69. - 91 - 101 - 93 - 90
-211 - 201 - 175 - 130 - 112 - 212
- 75 - 66 - 106 - 66 - 77 - 110 - 195 - 294 (+ 16) - 277 - 91
(- 361)
- 134 - 87 - 34 - 20 - 31 - 57 - 41 - 57 - 88 - 68 - 170
4 r i m so - 109 - 106 - 102 - 100
- 69 - 70 - 65 - 61 - 57 - 118 -81 - 56
- 50 - 41 - 53 - 45 - 42 - 86 - 83 - 79 - 43 - 36 - 29
- 47 - 40 - 84 - 40 - 38 - 30 - 26 - 23 - 29 - 63 - 61 - 58
- 51 - 68 - 61 - 33 - 36 - 26 - 27 - 26 - 62 - 61 - 79
113
As,,", SP
-215 - 183 - 190 - 184
- 53 - 55 - 44 + 58 + 57 - 114 - 64 + 36
- 19 - 50 - 48 - 48 - 48 - 125 - 118 - 96 - 87 - 76 - 183
- 28 - 26 - 22 - 26 - 39 - 80 - 169 - 271
- 214 - 30
- 83 - 21 + 27 + 13 + 3
-31 - 14 - 31 - 26 -7
-91
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114 Y. Marcus and A . Loewenschuss
No. , ion Asi abs
1 14 PdCli- - 96 I16 RhC1;- - 157 118 PtCli- - 163 120 IrCI2- - 156 121 IrCIi- - 193 128 ReC1:- - 322
hydr. r/nm AS,",,,
hydr. 0.319' -27.1 0.33' - 60 0.3 13' - 27.4 0.335' - 26.4 0.34' - 59 0.324' - 26.9
AS:* 4fi,,,S? 4t,,s? hydr. - 42 - 56 + 14 - 70 - 82 + 12 - 109 - 58 - 5 1 - 103 - 54 - 49 - 107 - 80 - 27
(- 268) - 56
'Crystal ionic radius for monoatomic ions, from ref. 130, for co-ordination number 6, unless otherwise noted, thermochemical radius for polyatomic ions from the sources noted; 'from ref. 132; 'from ref. 133; dinter- polated value; 'from the mean ratio of r,, for this class of ions to the characteristic interatomic distance;'van der Waals radius according to E. J. King, J . Phys. Chem., 1970, 74,4590; gthermochemical radius from ref. 134; 'from ref. 24; 'Crystal ionic radius for co-ordination number 4 from ref. 132.
water were essentially zero (see Section 7). Thus the entire effect found for NaCl could be assigned to the C1- anion. On this basis the isotope effect on the transfer of the hydrogen ions could be estimated, and the absolute value s:bs(D+, D20) = - 23.8 J K - ' mol-' was established, relative to pabs(H+, H20) = - 22.2 J K-'mol--'.
6 Models Concerning the Entropy of Hydration Models of the ions hydrating in the gas phase and in an infinitely dilute aqueous solution have been used for the estimation of their entropies of hydration. Conversely, entropies of hydration of ions have been used to substantiate models of the ions in these two environments, in particular the latter, of course. Such models have implications beyond the measure of the entropy change. Some of these are merely empirical correlations between the entropies of the ions in the aqueous solution or the entropies of hydration with some physical properties of the ions, into which some physical meaning has been cast. Others arise from a more fundamental study of the issue. Empirical Correlations. -The 'model' of a monoatomic ion with an inert gas electronic configuration in the gas phase is extremely simple. Equations (10) to (13) for obtaining the standard molar partial entropy of such an ion in the aqueous phase can, therefore, be readily turned into empirical corre- lations of its entropy of hydration. This is accomplished by the subtrac- tion of 1.5R In Mri = 12.4715 In M,JK-'mol-' and of the constant 108.85 J K-' mol-' (for 298.15 K), according to equation (5) . A subtraction of the actual value of Sy(g) of an oxyanion from equations (14) and (16) converts them into empirical correlations of its entropy of hydration. Similar procedures apply to equation (18) for oxycations and equation (17) for complex ions.
Other empirical correlations of the standard partial molar entropies of aqueous ions were also proposed. Zolotarev'a pointed out that soconv(aq) is linear with the mobility uo of the ion in the aqueous solution at infinite dilution or with the reciprocal of its Stokes radius rs:
E. K. Zolotarev, Zh. Fiz. Khim., 1965, 39, 1075
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Standard Entropies of Hydration of Ions 115
S&,(aq) = kup + b = (kFe/6nqH2,)zf/r; + b (37)
where k and b are ion-independent constants and y~~~~ is the viscosity of water. In a series of further papers Zolotarev and his co -worke r~’~~ showed that the entropy of hydration of monoatomic cations is linear with the square root of the ratio of the ionization potential 4 to the radius r,
AhydrS? = * (J/ri)’’2 + B ( z ~ ) (38)
The values of the constants A and B depend on the charge of the ion, and equation (38) is applicable not only to cations with inert gas electron con- figuration but also to transition-metal cations. The recalculated values of the constants (for I , in V, r, in nm) are: for z, = 1, A = - 21, B = 98; for z, = 2, A = - 16, B = 43; for z, = 3, A = - 11, B = - 115; to give AhydrSY in J K-’ mol-’.
Krestovlo6 showed that the standard molar entropies of the trivalent lanthanide, Ce4+, ad the tri- and tetra-valent actinide cations are linear with their enthalpies of hydration, divided by their charges. In a later study, Krestovlo6 developed this correlation further and extended it to other kinds of ions, including polyatomic ones. He showed that for two ions i and j of similar structure and charge,
AhydrS: - A h y d r q o = - 0*615 (S~%ns - ’!$%ins)
+ 0.345 Iz,I-’(Ahydr@ - A h y d r q o ) (39)
Empirical correlation expressions for the standard molar entropy of hydration were proposed more recently by Posin and Abramzon.lo7 For monoatomic ions the expression is
AhydrSl:onv = 121 - 41.7[1z11 - ((f/nm) + + 14zlJK-’mol-l (40)
where rp is the Goldschmidt crystal ionic radius and K’ = 0.028nm for cations and -0.028nm for anions. For polyatomic ions the expression is
AhydrSl:,,nv = 42 - 7.41[1z,1 - {(r:h/nm) + + 1 4 ~ ~ J K - l m o l - l (41)
with the same values of A”, and where r:h is Yatsimirskii’s thermochemical radius. The original expressions provided the absolute standard molar entropy of hydration on the basis of S&(H+, aq) = - 3.4cal K-’ mol-’ (1 cal = 4.184 J), and the last term in equations (40) and (41) converts the values to the conventional scale with S&v(H+, aq) = 0. Modifications of the Born Equation. - If a large charged sphere is transferred from the gas phase into water, the change in entropy involved is given by the negative of the temperature derivative of the electrostatic work done. This
lo’ E. K. Zolotarev and V. E. Kalinina, Zh. Neorg. Khim., 1962, 7 , 1225; Trudy PO Khim. i Khim. Tekhnol., 1963, 207; V. G. Gorelov, E. K. Zolotarev, and A. M. Yudin, ibid., 1964, 383; E. K. Zolotarev, Zh. Fir. Khim., 1965, 39, 884.
Io6 G. A. Krestov, Radiokhimiya, 1963, 5 , 258; Zh. Fiz. Khim., 1965, 39, 823. lo’ L. M. Posh and A. A. Abramzon, Zh. Obshch. Khim., 1976, 46, 2326.
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116 Y. Marcus and A . Loewenschuss
holds, provided that the temperature coefficient of the surface potential is taken into account, or if an equivalent charge of the opposite sign is trans- ferred simultaneously. For the single charge, the electrostatic work is given by the Born equation,13 hence the entropy change per mole of charged spheres transferred is
Ahyd, S,",,,, = (NAe2z,'/87r~,)r; I E - ~ ( & / ~ T ) ~ (42)
where zi is the charge of the sphere, Y, its radius, and E is the relative permittivity of the water, being the permittivity of vacuum. Spheres of smaller radii may now be considered, of sizes commensurate with those of the ions dealt with in this review. It is then found that application of equation (42) to a cation and an anion and summation of the resulting values does not produce a sum that agrees with the experimental entropy of hydration of the corresponding electrolyte (the sum of the conventional entropies of hydration of these ions). This failure of the Born equation to reproduce experimental entropies of hydration has led to various attempts at its modifi- cation. Two obvious modifications take cognisance of the change in the volume at the disposal of the ion on being transferred from its gaseous standard state to its acqueous one, and of the occurrence of dielectric saturation near the ion, so that the relative permittivity E is there much lower than in bulk water.
The change of the volume at the disposal of the ion leads to the so called 'compression term' of the entropy of hydration. For the standard states used in this review, the ideal gas at the pressure P = 0.1 MPa, where the volume is V = RT/P per mole of ions, and the (hypothetical ideal) 1 molar aqueous solution, where the volume is 1 dm3 per mole of ions, the compression term is
AcompSY = R In (1 dm3 - 0.1 MPa/RT) = - 26.7 J K - ' mol-' (43)
where the last equality is valid for 298.15 K. This must be added to the electrostatic term for the sake of comparison with the experimental entropies of hydration.
Some authors (Frank and Evans,"' Criss and co-w~rkers,' '~ Franks and Reid,"' and Criss and Salomon,"' for instance) chose the unit mole fraction standard state for the solute ion in solution. This was an unreasonable choice for a number of reasons. One is that if the ion is the only component in the liquid (xi = I), then it is impossible to add the contributions for the cation and the anion to obtain the value for the electrolyte (not to speak of the space charge created, that is not neutralized). Another reason is that no examina- tion of solvation (hydration) can take place if the ion is not embedded in the solvent, which is absent when xi = 1. It is understood, of course, that this standard state of xi = 1 was not considered as realizable by the authors who lo* H. S. Frank and M. W. Evans, J . Chem. Phys., 1945, 13, 507. Io9 C. M . Criss, R. P. Held, and E. Luksha, J . Phys. Chem., 1968, 72, 2970. ' l o F. Franks and D. S. Reid, J . Phys. Chem., 1969, 73, 3152. I " C. M. Criss and M. Salomon, in 'Physical Chemistry of Organic Solvent Systems', ed. A. K. Covington
and T. Dickinson, Plenum Press, London, 1973.
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Standard Entropies of Hydration of Ions 117
chose it, but was attained by extrapolation. It is, however, much more reasonable to use extrapolation to the reference state of infinite dilution in the aqueous solution, with the standard concentration of 1 molar. Infinite dilu- tion is the state where the solute-solvent interactions, which are the subject of this review, are best studied, and where the additivity rule for cationic and anionic contributions holds. Further reasons for rejecting the unit mole fraction standard state, based on statistical thermodynamic arguments, were given by Ben-Naim."* Consequently, the claim by Criss and Salomon,"' that the common concept that some ions in water disorder the solvent is an artefact of the standard state, is unfounded. The disordering effect (see Section 7) is arrived at with the use of the compression term, or another 'neutral term' (see below) that compensates exactly for the standard state effects in the experimental, conventional standard molar ionic entropies of hydration.
Another objection to equation (43) could have been raised by those authors, such as Frank and Evans,'" who advocated the use of the 'free volume' for the ion in the solution, instead of the total volume of the solution that contained 1 mole of the solute ion. The argument was that most of the latter volume was occupied by the water molecues, and only a small fraction of it (0.36% in water, calculated according to the method of Frank'I3) was at the disposal of the ion. The concept of a 'free volume' has, however, been abandoned since that time, it being realized that the ion, or any solute for that matter, has its full complement of translational degrees of freedom, pertain- ing to the total volume of the solution. Friedman and Krishnan,' who adopted in general the approach of Frank and Evans,"' did not accept their calculated value for the compression term, and used - 25 J K-' mol-', instead of equation (43), as a rough estimate.
The high electrostatic field near the surface of the ion, of the order of 109Vm-', causes dielectric saturation of the solvent in this region. This is manifested in a drastic lowering of the relative permittivity of the solvent, down to the limit of the electronic polarization (n2, where n is the refractive index), since all the dipoles of the solvent are permanently oriented by the field of the ion. At increasing distance from the ion the electric field strength weakens, and the relative permittivity assumes values between n2 and its value in the bulk solvent, E . Another manifestation of the high field near the ion is the compression of the solvent, its electrostriction.
The electrostriction of the solvent is a recognizable effect of the placement of a charged particle in it. From this arises114 an expression of the entropy of hydration as Ahydr S: = Ah,,& SSorn + Aelstr Sy . However, these two terms are not independent, since the former implies the latter. The volume change of the solution due to electrostriction can, in principle, be derived from the entropy of hydration:
Aelstr Y o = - AhydrSiO (a&laP)..I(a&laT),E (44)
' I 2 A. Ben-Naim, J . Phys. Chem., 1978, 82, 972. ' I 3 H. S. Frank, J . Chem. Phys., 1945, 13, 493. ' I 4 B. E. Conway, R. E. Verall, and J . E. Desnoyers, Z . Phys. Chem. (Leipzig), 1965, 230, 157; see also
J. Padova. J . Chem. Phys., 1963, 39, 1552.
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Since, however, the field strength E depends on the distance from the ion and on the local relative permittivity, E(r), the partial derivatives in equation (44), that have to be obtained at constant E, are not known, Hence AhydrS: cannot be obtained from measurements of partial molar volumes, that yield a value for the volume change of electrostriction. 'I4
Another often suggested modification of the Born equation is the recog- nition that also uncharged molecules, when transferred from the gas phase to the aqueous solution, have a standard molar entropy change that is not given by the compression term alone, equation (43), but require an additional 'neutral' term. Such considerations, together with an empirical expression as a power series in the ionic radius, led to the expression proposed by Noyes86
AhydrSY = 619(rp/nm)2 - 4.067zf(r:/nm)-' - B(r:/nm)-2 - 30.7 (45)
for the entropy in J K-' mol-' at 298.15 K, where rp is the Pauling crystal ionic radius, and B = 0.167 for cations and B = 1.977 for anions. The first term represents the 'neutral' term, and second one the Born electrostatic term according to equation (42), the third one an empirical extension, and the fourth the compression term according to equation (43) (but more negative). Equation (45) was supposed to yield the absolute standard molar entropy of hydration of the ion, and was tested for univalent ions of both signs.86 It yielded too low a value of - SPabs(H+, as), however, only 4.9 J K-Imol-', compared with the now accepted value of 22.2 J K-' mol-' (see Section 5) .
A recent revival of these ideas is due to Abraham, Liszi, and co-workers,''5 who developed a multilayer solvation (hydration) model. The entropy of hydration is the sum of a neutral term and an electrostatic one. The former includes the compression entropy according to equation (43) and is
s,:,,, = - 53.7 + 439(r,,/nm) (46)
where ric is the crystal ionic radius of the ion. Equation (46) was obtained as an average of the entropies of solution of gases with molecules of different sizes in water. The electrostatic entropy term, according to the multilayer concept, is
Si:l = - (N,e2z'/8neo)[~12(as,/aT)(b-' - a-I)
+ &2-Z(a&2/aT)(c-' - b-1) + &;*(a&b/aT)c-I]
where is the relative permittivity of the first hydration shell, E* that of the second shell, and &b that of bulk water. The distance a is identified with the crystal ionic radius, ric, the distance b with a + rw, where rw = 0.147 nm is the thickness of the first hydration shell, and the distance c is that beyond which the water has its bulk properties. If the ion is so large that it has no hydration layers, then c = b = a = ric, and equation (47) collapses into the Born equation (42). If dielectric saturation is assumed to occur in the first layer, then = n2 should hold, where n is the index of refraction of water,
(47)
'I5 M. H. Abraham and J . Liszi, J . Chem. Soc., Furuduy Trans. I, 1978,74, 2858; M . Abraham, J . Liszi, and L. Meszaros, J . Chem. Phys., 1979, 70, 2491.
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Standard Entropies of Hydration of Ions 119
but the empirical quantities = 1.87 and (ae,/aT), = 2.6 lOP4K-' were deemed to give better agreement (for a one layer model, i.e., one in which c = b). The relative permittivity c2 in the second layer and its temperature coefficient could not be estimated from independent data, however; the term - ( N , e 2 z ~ / 8 n ~ , ) E ; ~ ( ~ E , / ~ T ) , ( C - ' - b- ' ) was, therefore, left in its entirety as a fitting to the experimental quantity, being then given a structural inter- pretation (see Section 7). This particular difficulty could be circumvented by setting c = b, i.e., by the recognition of only one hydration layer. The one-layer model was extensively tested for nonaqueous solvents, but for water, so it seems, the one-layer model is inadequate. An extension of the study to higher temperatures, up to 527 K, showed that beyond 400 K all the ions are structure enhancing. Below 423 K some ions are structure-breaking, but the structural effects cannot be calculated from the electrostatic Models Based on Cavity Formation - An alternative to the use of experimental data on gas solubilities for the estimation of the neutral term, Asgut, is its estimation from the scaled particle theory.'I7 Arakawa et aL"' and Sen''' employed this theory for the calculation of the Gibbs free energy and the enthalpy changes occurring when a cavity of a specified size is created in a solvent (water) for the accommodation of a dissolving particle (ion). From these quantities the entropy change for cavity creation, AcavSY = (AcavH? - Acav GP)/T is directly obtained. The input quantities for the scaled particle theory calculations are the molar volume of water, VH20(T), and its isobaric expansibiity C I ~ ~ ~ ~ ( T ) for the temperatures at which the theory is to be applied: 273 to 343 K"' or -c 373 K and 373 to 643 K."' Further required are the diameter of a water molecule regarded as a hard sphere ( Q H , ~ = 0.282 nm)"' or as a temperature-dependent quantity,"' and the diameter of the ion for which the cavity is required, taken as twice the (Pauling) crystal ionic radius, 2rp, a temperature-independent quantity.
The contribution from the entropy of cavity formation, AcavSio, is tem- perature-dependent (even if the diameter of the water molecule is considered as constant) because of its dependence on the packing of the water molecules in the liquid. It is a sizable fraction of the entropy of hydration only at relatively low temperatures,' '*J" but loses its importance as the temperature increases, and becomes neglible at T > 450 K.'19
Models Based on Co-ordination. - The disagreement of the value of AhydrSO, calculated as the sum Acorn S," + Acav S," + ASSorn, with experimental values noted in the work of Sen!" shows that other sources of entropy are import- ant in the hydration process. Not least among them is that rising from the direct interaction between the ion and the water molecules surrounding it. This was included in the multilayer electrostatic approach of Abraham and
'I6 M. H. Abraham and J. Liszi, J. Chem. Soc., Faraday Trans. I, 1980, 76, 1219; M. H. Abraham, E. Matheoli, and J . Liszi, ibid, 1983, 79, 2781. H. Reiss, H. L. Frisch, and J. L. Lebovitz, J. Chem. Phys., 1959, 31, 369; R. A. Pierotti, Chem. Rev., 1976, 76, 717.
'Ix K. Arakawa, K. Tokiwano, N. Ohtomo, and H. Uedaira, Bull. Chem. Sor. Jpn., 1979, 52, 2483. U. Sen, J . Am. Chem. Soc.: 1980, 102, 2181; J. Chem. Soc., Faraday Trans. I, 1981, 77, 2883.
117
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120 Y. Marcus and A . Loewenschuss
in the dielectric saturation of the first hydration shell, leading to values of and (&,/aT), that differ considerably from those of bulk water. More direct evaluation of the entropic contributions of the co-ordination of water molecules with the ion in this shell have also been proposed by other authors.
Bockris and Saluja12' considered various models for the state of the water molecules in the first two concentric solvation shells around an univalent monoatomic ion. Although none of the models considered yielded values of the entropy of hydration in agreement with experimental values, the least inconsistent model is the following one. The first solvation shell contains some water molecules that are structurally co-ordinated to the ion (SCW) and others that are not (NSCW), but only juxtaposed to it geometrically. The total number of water molecules in the first solvation shell is obtained from X-ray diffraction. In the second shell the water molecules are all monomeric, i.e., not bonded to others as in the normal tetrahedral structure that prevails in bulk water. Three kinds of interaction are involved in this model, each yielding its own entropic effects through modifications of the degrees of freeciom of rotation, libration, and vibration of the water molecules: Sscw-, , SNsCw-,, and SwPw , where W and i in the subscripts designate water molecules and the ion, The entropy of hydration is then given by
~ i ~ ~ i 1 1 5,116
A h y d r S P = [S;rans(aq) - sP(g)l + ~ s P , o r n
+ ~SCW(S,CW-, - S w - W ) + ~ N S W ( S N S C W - I - &-w) + Z A s b r e a , (48)
where n is the number of water molecules of the kind specified by the subscript, and XASbreak is the entropic effect of the water structure that is broken in the second hydration shell. This treatment yielded the surprising raul t of zero for the translational entropy of the ion in the solution, S,,,,,(aq). As mentioned above, the results agree poorly with the exper- imental entropies tested, except for the fortuitiously well agreeing value for Rb+ .
Goldman and Bates121 considered a model where the ion X' is co-ordinated with n water molecules and undergoes the following thermodynamic cycle:
WIC, 8) + nH,O(g) -L XZ(H20)n(rlc, g)
(49) t t 2 1 4 *
xz(rlg7 g) + nH20(1) hydr ' X'(H20)"(rlC9
The entropy change for the hydration process, the lower line of the cycle, is the sum of those of the four steps around its upper part. Now, A 1 S P = 0, since it is immaterial for the entropy whether the gaseous ion has its radius T , ~ = Y,, or whether its sizes differ in the two phases, the radius in the aqueous solution taken as equal to the radius in crystals. The literature value3 of AvapS&O is used in step 2 to give A2S: = nAvap Sg20, the value of n considered being limited arbitrarily to 3,4, 5 , and 6. The values of A3S: is calculated for
IZo J . O'M. Bockris and P. P. S. Saluja, J . Phys. Chem., 1972, 76, 2291. ''I S. Goldman and R. G . Bates, J . Am. Chem. SOC., 1972, 94, 1476.
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Standard Entropies of Hydration of Ions 121
symmetrical configurations with n = 3 (equilateral triangle), 4 (regular tetra- hedron), and 6 (regular octahedron). The arbitrary choice of n depends on the ion. The calculation takes into account the change, when n + 1 particles form one particle, in the translational entropy (including the change in mass) as well as in the rotational entropy. This is why only symmetrical structures are considered. The water molecules are permitted to rotate around their dipole axes, which point towards the ion. The change in the vibrational entropy in this step of the cycle involves the fundamental breathing frequency of the water molecules and oscillations around their major and minor axes of inertia. The value of A3S: for n = 5 is interpolated between those for n = 4 and n = 6. Finally, AqSO includes a Born term, equation (42), the term (288/T)zfr,k, where for both terms the effective radius is obtained from the volume of the hydrated ion as rieff = ( r i + nr&o)i, with Y ~ , ~ = 0.126nm, and also a combined neutral and compression term.
This method involves detailed calculations of the entropic contributions of the various degrees of freedom changes in the gas phase co-ordination of water molecules to the ion, step 3. These could have been replaced by the experimental results discussed in Section 4. If this were done, then a test of the notion that co-ordination with n water molecules according to equation (49) completely accounts for AhudrSO would have been feasible. As it is, the calculations involve many assumptions concerning the rotational, libra- tional, and vibrational states of the co-ordinated water molecules, that are not verifiable independently.
Use of the gas-phase ion hydration data discussed in Section 4 was, in fact, made by Gonzalez Maroto et a1.’22 They employed a cycle similar to equation (49), and took the values for A3S0 from K e b a ~ l e ’ ~ ~ for the alkali metal and halide ions. For A4SY they added four contributions: the .change in the polarization of the solvent around the ion, the entropy of mixing to form a 1 mol dmP3 solution, a translational entropy change, and a structural entropy change. They did not preselect a value for n, however, but presented the total entropy of hydration [the bottom line in cycle (49)], calculated as a function of this number of water molecules added, n, and the co-ordination number, N,, of the ion. From a similar calculation of the enthalpy of hydration they then obtained the Gibbs free energy change, and found that for 5 6 N, < 8 it did not depend appreciably on the co-ordination number. With increasing n the values of AhydrGy were found to decrease continuously, except for Li+, Na+, K+, and F-, for which a shallow minimum was reached at n = 3 or 4. The main effect of the co-ordination number comes into the structural entropy term, as N, - 2n (for cations) or N, - 3n (for anions) hydrogen bonds ruptured or formed when gaseous hydrated ions are introduced into water in step 4 of the cycle (49). The following pairs of (N,, n) values yield calculated AhydrSi: values that correspond most closely to the experimental AhydrS& data that are listed in Table 5: Li+(5, 5), Na+(6, 6 or 7, 7), K+ and
‘22 R. Gonzalez Maroto, D. Posadas, M. I. Sosa, and A. J. Arvia, An. Asoc. Quim. Argentinas, 1982, 70, 979. P. Kebarle, in ‘Modern Aspects of Electrochemistry’, ed. B. E. Conway and J. O’M. Brockris, Plenum, New York, N.Y., 1974, Vol. 9.
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122 Y. Marcus and A . Loewenschuss
Rb+(5, 4 or 6, 3, Cs'(6, 5) , F-(6, 6), C1-(7, 4), Br-(7, 3), and 1-(8, 4). There are several unverifiable and even downright unreasonable assumptions in this treatment, e.g., the setting of the effective radius of an ion as re = (r;c + nr3,20)1/3. Hence the significance of the (Nc, n) pairs listed above is questionable, although the numbers are quite reasonable. The main merit of this approach is the use of the experimental gas-phase entropies of hydra- tion, that reduces the problem to the transfer of the hydrated ion from the gas phase into the solution, with the single additional free parameter, n.
The recent attempt by R y a b ~ k h i n ' ~ ~ in the same direction is more modest, since it incorporates certain experimental data without trying to calculate them from more fundamental premises. The expression for the entropy of hydration (for the monoatomic ions considered) that is derived from this work is
where the terms have the following meanings and values. The change in the translational entropy involves the change in mass, when n water molecules co-ordinate to the ion: AS&,, = 1.5R In [(Mri + nMrH20)/M,]. The next three terms pertain to the entropy changes of the melting of n moles of ice, ASf, heating them from 273.15 to 298.15 K, ASn, and the loss of three degrees of freedom of orientation per molecule in n moles of water. The change in rotational entropy depends on the symmetry of the ion, that is co-ordinated to n molecules of water: AS:,, = 1.5R In T - R In Q ~ ( ~ ~ ~ ) , / Q ~ ~ ~ , where Q is the symmetry number. The interaction term is proportional to (2: + 1)1'2 x [(ric/nm) + 0.057]-', and the last term is positive for anions and negative for cations, but of the same absolute magnitude. Ryabukhin claimed that setting n = 6 or n = 8 for different ions yielded agreement between the standard molar entropy calculated from his equation with experimental values. Conclusion. -The models presented in this Section demonstrate that the hydration of an ion is an extremely complicated process, as far as the standard molar entropy change is involved. Frank and Evans"* were among the first to analyze the entropy of hydration of ions in terms of the following contributions, although not all of them were named by them in the same manner. (i) A compression term, from the standard state in the gas phase to that in solution, equation (43). (ii) A neutral term, that may be approximated by the entropy of solution of inert gases of appropriate sizes, or by the entropy of cavity formation, or otherwise, Each of these modes may, however, involve aspects covered also by other items of this list. (iii) A long-range electrostatic term, that operates beyond a number (one or two) of hydration shells or a distance to be specified, according to the Born equation (42), the specification yielding the needed value of ri. ( iv) An immobilization term, that describes the co-ordination of a specified number of water molecules to the ion, so that they move in the solution
A. G. Ryabukhin, Zh. Fiz. Khim., 1981, 55, 1670.
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Standard Entropies of Hydration of Ions 123
together with it rather than freely. Changes in the rotational and vibrational degrees of freedom of these water molecules and of the ion, if polyatomic, also due to this co-ordination and have entropic effects. (v) A term describing changes in the structure of the water situated between the inner hydration shells, where the water is coordinated and the entropic effects are covered by item (iv), and the bulk water.
Frank and Evans"' lumped items ( i ) and (ii) together and discussed them in terms of the concept of a free volume, and added the separate contri- butions of items (i i i) , (iv), and (v) . Items (i) and (iii) constitute contributions which are accepted by most authors and incorporated in one way or another into their schemes. This Section dealt maily with items (ii) and (iv), about which little agreement prevails. This is true also concerning the question of how the various effects can be disentangled from one another, so as not to consider the same effect more than once, under its different aspects. The last item, ( v ) , is the subject of Section 7.
7 Water Structure Effects and the Entropy of Hydration of Ions
Water Structure and Entropy. -The qualitative notion of the structure of liquid water is well established, although there is no general agreement regarding its quantitative expression (for a recent contribution to this ques- tion see Marcus and Ben-Naiml"). It is also well known that ions affect this structure, some being water-structure-makers, i.e., they enhance the inherent structure, and others are water-structure-breakers, i.e., they weaken it. The former group includes most multivalent ions and some small univalent ones, the latter group includes large univalent ions (but not hydrophobic ones!) and very few very large multivalent ions (see Engel and Hertz'26). Water- structure- making seems to be the rule among the ions, and its breaking is the exception, but the latter is well documented and accepted as a fact, the apparent statement to the contrary by Criss and Salomon"' notwithstanding. It is manifested in many experimentally observed phenomena, where it is the only straightforward explanation. For instance, the fluidity of a solution containing certain ions is enhanced (the viscosity is lowered), as pointed out by several author^.'^ This can be expressed as a positive value of the B-coefficient of the Jones-Dole viscosity equation,lZ7 and as emphasized by Gurney84 is ascribable to the water-structure-breaking properties of these ions. Water-structure-breaking by ions is also manifested as 'negative hydra- tion', a concept introduced by Samoilov'" to express a negative value of the difference between the energy of activation for the exchange of water molecules from the vicinity of the ion and that for bulk water.
'3 Y. Marcus and A. Ben-Naim, J. Chem. Phys., 1985, 82, in the press. i26 G. Engel and H. G. Hertz, Ber. Bunsenges. Phys. Chem., 1968, 72, 808. '*' G. Jones and M. Dole, J. Am. Chem. SOC., 1929, 51, 2950. 12' 0. Ya. Samoilov, 'The Structure of Electrolyte Solutions and the Hydration of Ions', Izd. Akad. Nauk
SSSR, Moscow, 1957; Consultants Bureau, New York, NY, 1965.
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124 Y. Marcus and A . Loewenschuss
The ratio of the longitudinal relaxation rate of water molecules in the presence of an ion, t l l , to that in its absence, t lo , was given by Engel and Hertz'26 in an expression of the type of that of Jones and Dole for the viscosity:127 ( t ; ' / t ; ' ) = 1 + B:c, + . . . The ratio of the rotational corre- lation time of a water molecule in the presence of an ion, zI , to that in its absence, z,, is then given by the expression z,/z, = 1 + (n, VH20)-'B:, where n, is the number of water molecules in the immediate vicinity of the ion (in the first hydration shell) and VHzo = 0.01807dm3mol-' is the molar volume of water (for c, being the concentration on the molar scale). It was found126 that there are ions for which B: < 0 [on the basis that B'(K+) = B'(C1-)], i.e., z,/T, < 1 . This signifies that these ions are water- structure-breaking, since the relatively large value of z, is predicated on the hydrogen-bonded structure of water.
There are several more properties of aqueous electrolyte solutions that show that for some ions, the water-structure-breaking ones, there is a 'thawed' zone of water around the ions in the sense of Frank and Evans'" and of Frank and Wen ( 1957).129 In such a zone the water is less 'ordered' and participates in less regular hydrogen bonding that in bulk water. Water- structure-breaking should, therefore, also manifest itself in a positive contri- bution to the entropy of hydration of these ions. Perusal of the conventional standard molar entropies of hydration of ions in Table 1 , and of the absolute standard molar entropies of hydration of ions in Table 5 , shows all of them to be negative. This is true even for the ions classified as water-structure- breakers according to the methods mentioned above. There must, therefore, be some negative contributions to the entropy of hydration that predominate aver the structure-effects of such ions. These contributions have been dis- cussed in Section 6, and listed in its concluding remarks. It was mentioned there that general agreement exists over the compression and long-range electrostatic terms, both negative contributions to the standard entropy of hydration. Their subtraction from the absolute standard molar entropy of hydration leaves a remainder
Ahydrs?* = AhydrS%s - AcompS? - AhydrSzorn(rI) (51)
that is marked here with an asterisk but can remain nameless. It should be better able to show the positive entropic effects of water-structure-breaking ions, and is listed in Table 5. Indeed, the Table shows (very) few cases where A h y d r s 7 * is positive. That it is not positive for many other water-structure- breaking ions is due to another effect, also discussed in Section 6, of the immobilization of water molecules in the vicinity of the ion, an effect that is elaborated on further below. In fact, the entropy change for the effects of ions on the structure of the water, AStrUCS,O, and that for the immobilization of the water molecules near the ion, Atnm S,O, together make up the value of A h y d , s p *
[see equation (56) below].
'29 H. S. Frank and W.-Y. Wen, Disc. Furuday Sac., 1957, 24, 133.
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Standard Entropies of Hydration of Ions 125
A Calculation of AstrucSi .-At this stage it is necessary to specify the value of ri in equation (42) in order to make a concrete application of equation (51). There is general agreement that the Born equation that employs the relative permittivity of the bulk water and its temperature coefficient should not be applicable nearer to the ion than beyond its first hydration shell. There is no agreement, however, on any given larger distance, whether beyond the second hydration shell or otherwise. Following Frank and Evans and others, for the present purpose
r,(Born) = rlC + 0.28nm (52)
is used, where rlc is the crystal ionic radius and 0.28 nm is the diameter of a water molecule. This uncertainty in r, may mix-in electrostatic effects that should be expressible by the Born equation with an unknown but more appropriate value of rl into the structural and immobilization effects, that are to be disentangled once AhydrSP* has been calculated.
Values of ric are available for most monoatomic ions from Shannon and Prewitt. 130 These are, essentially, Pauling-type radii for co-ordination number 6, that reporduce the cation-anion distance in most fluoride and oxide crystal structures with the specified radii rlc(F-) = 0.133nm and r l c ( 0 2 - ) = 0.140 nm. They are in substantial agreement (within 0.005 nm) with the original PaulingI3' radii and with their revision by A h r e n ~ . ' ~ ~ In view of the addition of 0.28nm to give ri(Born), the small discrepancies between these scales are immaterial. The Pauling crystal ionic radii or revised values thereof were preferred over other scales (Goldschmidt or Gourary-Adrian) since the former seem to conform best to the radii of the aqueous ions, as established by Marcus.'@
Values of Tic for polyatomic ions are less readily available. The therrno- chemical radii of Yat~imirski i '~~ and Kap~s t in sk i i ' ~~ are widely used for purposes similar to the present one. These radii were reviewed, amended, and supplemented by Jenkins and T h a k ~ r , ' ~ ~ but there are still many ions in Tables 1 and 5 for which no radii are available from these sources. The polyatomic ions were, therefore, classified into groups of similar struc- ture, and for each group the mean of the values of the ratio of the thermo- chemical radii, where available, to the characteristic interatomic distance was calculated. This distance, e.g., the C1-0 distance in ClO,, was that used for obtaining the rotational contribution of So(g) of the The means found are: 1.32 0.10 for the diatomic ions, 1.60 0.04 for triatomic ones (whether linear or bent), 1.33 0.07 for tetraatomic ones (whether planar or pyramidal), 1.53 f 0.03 for tetreahedral ones, and 1.41 & 0.04 for octa-
13' R. D. Shannon and C. T. Prewitt, Acta Cryst., Sect. B, 1969, 25, 925; 1970, 26, 1046. 13' L. Pauling, J . Am. Chem. SOC., 1927,49,765; 'The nature of the Chemical Bond', Cornell University Press,
1 3 2 L. H . Ahrens, Geochim, Cosmochim. Acta, 1952, 2, 155. '33 K. B. Yatsimirskii, Izvest. Akad. Nauk SSSR, Otdel. Khim. Nauk., 1947, 453; 1948, 398. '34 A. F. Kapustinskii and K. B. Yatsimirskii, Zh. Obshch. Khim., 1949,19,2191; A. F. Kapustinskii, Quart.
'35 H. D. B. Jenkins and K. P. Thakur, J . Chem. Educ., 1979, 56, 576.
Ithaca, NY, 3rd Ed., 1960.
Rev. (London), 1956, 10, 283.
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126 Y. Marcus and A . Loewenschuss
hedral ones. These means were used here for the estimation of ric values for other ions in these groups.
The value of the radii ric, whether reported or estimated, are shown in Table 5 , and were used to calculate AhydrS$orn from equation (42) with the ri values obtained via equation (52). These calculated electrostatic contri- butions to the entropy of hydration are also presented in Table 5. It is seen that for univalent ions A, drSgorn varies from - 12.0JK-'mol-I for the
range mainly around - 8 k 1 J K--' mol-'. An uncertainty of 0.02 nm in ric causes a maximal uncertaity of f0 .7JK- 'mol- ' in AhydrS:Born for uni- valent ions. For the divalent ions the electrostatic contribution to the entropy of hydration ranges from - 53 J K-' mol-' for the smallest (Be2+) to - 26.4 JK-Irno1-l for the largest (IrCli-), most of the values ranging about - 38 k 8 J K- ' mo1-'. Here an uncertainty of k 0.02 nm in ric of the smaller polyatomic ions causes an uncertainty of k 1.8 J K- ' mol-' in the electro- static contribution. For the trivalent ions this uncertainty is even larger, k 3 J K-' mo1-'. All of these uncertainties are of the same order as 1 .4zi due to the conversion to the absolute partial molar entropies, and come on top of this and the other uncertainties noted in the compilation of Table 1.
The values of the electrostatic contribution to the standard molar entropy of hydration, Ahydrs:Born, together with the change in the entropy due to the change of the standard state, AcompSP = -26.7JK-'mol-', are used in equation ( 5 1) to obtain the values of Ahydr S:* listed in Table 5 . The cumula- tive uncertainties in the latter quantity are estimated at k 3zi. The values of Ahydrsy* of the monoatomic ions are plotted in Figure 1 against the reciprocal of the crystal ionic radius ric. It is seen that the values tend to group according to the (absolute) values of the charges and that within any charge group there exists a correlation with the size.
For the lack of a simple, yet compelling, theory for the divison of AhydrS;*
between the solvent immobilization effect and the effect of the ion on the structure of the water, the following simple approach was devised. Its purpose was to bring out the structure-breaking effects of those ions that exhibit them, as obtained also from other method^.^^,'^^,'^^ It should, therefore, not be taken too seriously in its quantitative aspects.
The notion that the water molecules in the near vicinity of the ions lack translational freedom but move together with the ion was borrowed from several authors. These gave it various expressions, from complete freezing, according to U l i ~ h ' , ~ and R y a b ~ k h i n , ' ~ ~ via 'half-freezing', according to Frank and EvansIo8 and Friedman and Krishnan,' to association with the ion, so that the number of particles having translational degrees of freedom and their masses are affected, according to Goldman and Bates.I2' If, then, hydration in the aqueous solution involves the reaction
smallest (Li+) to - 6.2 J K- T mol-' for the largest [(C,H,),N+], but that they
Xz(aq) + nH,O(I) - X(H,O)i (53)
'36 H. Ulich, 2. Electrochem., 1930, 36, 491.
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Standard Entropies of Hydration of Ions
I I I
Br cs Rb 0 TI
0 0 0
O K 0 s No
:g 0
OSe Ro - 0 O F 0 c u
80 Pb 0 0
Eu Oosr Hg
Co
Sn
00
0 0 0 P+ FoCU oCr oMg
O B I OC0 ONi
"0 Gd
Pr OY DyO oYb
0 Er
Pu
u o B ~ 0
Thoo -=.- 0 Hf
0 Zf
Li 0
OSC
Cr
OGo
v 9 x 0 Fe
A
127
I I -600' 5 hi, / nrn)- '
Figure 1 Values of Ahydr@*/J K- ' mol-'for monoatomic ions (0 cations; 0 anions) obtained by equation (51) from the data in Table 5 , against the reciprocal of the crystal ionic radii, also listed in that Table
the mass of the ion is increased from that of x' to that of X(H,O)E and of the n + 1 particles only one has translational freedom. The translational entropy of the water molecules in liquid water is taken to be
StF),,s(H20, 1) = SO(H,O, 1) - [S0(H,07 8) - StS)ans(H*Ol &I = 26.0JK-'mol-' at 298.15K (54)
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128 Y. Marcus and A . Loewenschuss
on the assumption that the non-translational degrees of freedom of the water molecules are not affected. The values of the standard molar entropy of liquid and gaseous water are from the NBS table^,^ and the translational entropy of gaseous water is calculated by equation (5). The loss of translational entropy when ni moles of water are immobilized near a mole of ions i = X' as in equation (53) is therefore
AtrImS: = 1.5R In [Mr{X(H,O),J/M,(X)] - 26.0n, (55)
The number n, of water molecules so affected is etimated by taking into account the trends noted in Figure 1, on the one hand, and the presumed indifference of the sodium ion with regard to water-structure-making or -breaking5,Io3 on the other. Thus ni is taken to be proportional to )ziI/riC, the proportionality constant being fixed so that for the sodium ion At,im:;iO(Na+) from equation (55) equals the value of dhydrSp*(Na+) in Table 5. The result is
n, = 0.3551z,I/(r,,/nm) (56)
The water-structure-affecting properties of the ions are thus expressed in terms of the following structural entropy change
~ s t r u c s P = AhydrSP* - Atrim SP (57)
The values of AtrimSp and AstructS: calculated from equations (55) and (57), respectively, are shown in Table 5.
It is conceded that equation (55) and the manner through which the value of ni is obtained are extremely oversimplifed representations of the entropic effects concerned. This is shown clearly in the cases of the smallest mono- atomic ion of each charge-type examined: Li+, Be2+, and A13+, where ni becomes much larger than the hydration numbers expected for these ions. As a consequence AtfimS: of these ions result, which must be discounted. Never- theless, equation (55) and the manner of calculation of ni have the merit of great simplicity, on the one hand, and of a reasonable resulting list of ions with positive values of AStruCSY, on the other (with the above-noted excep- tions). This list is shown in Table 6, together with the results from several other lines of investigation, which are seen to be in adequate concordance.
Other Evaluations of AshcSi . -The evaluation of the structural entropy change from equations (51) and (57), i.e., from the difference between the absolute standard molar entropy of hydration and the sum of the contri- butions from the compression, electrostatic, and immobilization effects, was accepted by many authors since the work of Frank and Evans.'" Some previous interpretations of the entropy of hydration, however, disregarded the structural entropy change altogether, as may be seen from the discussion in Section 6. For instance, U l i ~ h ' ~ ~ recognized only the compression term in addition to the immobilization effect, and divided the difference AhydrSPabs - AcompS: by the entropy of translational immobilization per mole, to obtain the number ni of water molecules per ion that is so immobilized. He assumed the molar entropy of water immobilized in the hydration shells of ions in
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Standard Entropies of Hydration of Ions 129
Table 6 Water-structure-breaking ions (4- ) according to various approaches [some structure-making ions (-1 and indiferent ones (+> are included for comparison J
Ion
Li +
Na+ K + Rb+ c s +
Ag+ T1+ Ba2+ Ra2+ Sn2+ Pb2+ F- c1- Br- I- At- S2 - Se2- NH: (CH, )4N+ IC2 R, )4 N i 0; CN- NO; SCN- N; AgCl; NO, ClO, BrOy 10, BH; BF,- ClO; BrO; 10; MnO; TcO; ReO; CrO:- Cr&- so: - s,o; - S402- AuCl; AuBr; CdIi- HgIi-
Au(CN); Ag W ) ;
Hg(CN):-
entropy b
- + + + +
-
-
- + + +
+
viscosity exchange d e
n.m.r. f - - + + + -
-
- - + + +
- + + +
-
+ + +
+ -
+ + +
+ + + -
+ + +
+
+ + -k
+ + + -
+ + + -
+ + + + + +
+ + 4-
+ + - +
+ -
+ + + + +
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130 Y. Marcus and A . Loewenschuss
Ion entropy a b C
Co( CN); - -
PF, SnFi- + PdCli- + PtC1,2- + RhCli- +
viscosity exchange n.m.r. d e f
+
"This work, from the sign of AStmcSF, 'indifferent' (+) considered if - 3 < (-<,,,,SY/J K - ' mol-') < 3; the case of Li+ is discussed in the text; all the ions from Table 1 (or Table 5 ) not included in the present Table are structure-makers; bfrom ref. 5, according to the sign of Si ll (the entropy change of transfer of water from its pure liquid to the cosphere I1 about the ion); 'from ref. 138, according to the sign of <Sl1 [the sign may change from + to - , marked here, depending on the value of SEb, (H' , aq) used]; daccording to the sign of the (negative of the) Jones-Dole B-coefficient of the viscosity, on the assumption that B(K+) = B(C1-), with data from ref. 84, M. Kaminsky, Z . NaturS, Orsch., Ted A. , 1957,12,424, and other authors, summarized in ref. 84 and 126; eaccording to the sign of the (negative of the) activation energy E + for water exchange, from ref. 000 and other data summarized in ref. 128;/from the sign of the relative rotational correlation time of water, (T~ /T,,) - 1, according to ref. 126.
solution to equal that of water immobilized in crystal hydrates.The values available to him were (in J K-' mol-') 40 in sodium dihydrogenphosphate dodecahydrate, 47 in tetrapotassium hexacyanoferrate(I1) trihydrate, and 43 in oxalic acid dihydrate. Similarly, in ice, extrapolated to 298.15K, the entropy of the immobilized water is 41 J K-' mol-'. When he compared these values with the standard molar entropy of liquid water, 66.5 J K-' mol-' (the modern value is 66.91 JK-'mol-1),3 he ascribed the difference of -25 J K-' mol-' to the entropy change of the translational immobilization per mole of water. This figure is not far from the 26.0 J K-' mol-' employed in equation (55) . R y a b ~ k h i n ' ~ ~ used the value 28.6 J K-' mol-' for this quantity, but, contrary to U l i ~ h , ' ~ ~ also took the change in mass of the ion when it was hydrated into account, as in equation (55). Ryabukhin also allowed for the loss of orientational entropy of these water molecules, but this was compen- sated to a large extent by the gain in rotational entropy by the ion (a monoatomic ion with no rotational degrees of freedom being converted into a polyatomic hydrated one that had them).
Frank and Evans"* fixed the number of water molecules immobilized near an ion arbitrarily at ni = 4, and the amount of entropy lost per molecule of water at one half of what it would lose on freezing (at 298.15 K),136 i.e., 12.5 J K-' mol-'. Hence, for the univalent monoatomic ions considered, AtrimSY = - 50 J K-' mol-'. The authors mentioned, however, that this fig- ure may not be sufficiently negative for more highly charged ions. The only polyatomic ion considered by them, with no comment on it being so, was the ammonium cation, NH; .
That polyatomic ions constitute a special case was recognized by several authors. Nightingale'37 and Krestovlo6 noted that polyatomic ions may suffer a marked restriction of their rotation in solution, up to losing all their rotational entropy. This, however, seems to be too drastic a view. Although the rotational degrees of freedom of the 'bare' ion are modified by its association with the water molecules immobilized around it, as are those of
E. R. Nightingale, jun., J . Phys. Chem., 1959, 63, 1381.
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Standard Entropies of Hydration of Ions 131
these water molecules themselves, the hydrated ion can surely rotate. New degrees of freedom (libration, vibration) are thus created for the old ones that are lost. Only a very detailed analysis of this problem, along the lines taken by Goldman and Bates,12' can decide on whether rotational, libra- tional, and vibrational entropy together are lost or gained in the process of hydra tion.
The approach taken by Frank and Evans"' was adopted by Friedman and Krishnan,' who employed their value of A,,,Sp, but a corrected value for AhydrSSorn and different standard states (1 atm and 1 molal solution rather than unit mole fraction of solute), and therefore a different value of AcompSP. They called the structural contribution 'hydration of the second kind', and calculated it essentially via equations (51) and (57). Only monoatomic ions were treated by them, and the water-structure-breaking properties of the appropriate ions were brought out as positive values of the resulting A,, , , ,S~ .
In an extensive series of papers, published over nearly two decades, Kres to~ , '~ ' A b r o ~ i m o v , ' ~ ~ and their c o - w ~ r k e r s ' ~ ~ addressed themselves to this problem. It is noteworthy that this work was ignored by Friedman and Krishnan,' as far as what was published before their review went, and that on the other hand Kre~tov '~ ' disregarded the work of Frank and Evans"' and the previous work of Frank and Rob ins~n . '~ ' He complained that the litera- ture at the time appeared to contain no record of attempts to divide the entropy of hydration into two parts, one of which characterized the entropy change on formation of the aqueous ion, the other that of water acting as a solvent. It is precisely these two aspects that the works of Frank and co-workers mentioned above discussed. K r e ~ t o v ' ~ ~ also ignored the con- centric shell concept of Frank and Wen'29 but suggested the same model himself. The model of the hydrated ion consisted of four concentric shells, the fourth being bulk water, but the boundaries between the inner three shells could not be defined, so that they were lumped together, the model consider- ing in fact the 'bare' ion and its total hydration shell.
The absolute standard molar entropy of hydration of an ion was divided by Krestov as follows
AhydrSI% = As: + As:, (58)
where AS: pertains to the bare ion and ASP,, to the orientational and structural effects. The latter quantity was eventually ( A b r o s i m ~ v ' ~ ~ ) sub- divided into short-range (structural) and long-range (electrostatic) entropy effects, and an electrostatic contribution to AS,? was recognized, albeit only with regard to the temperature coefficient of the entropy of hydration (Krestov and K~benin'~'). The quantity AS,: was estimated empirically, on the basis of the entropies of solution of the inert gases in water. It was
'38 G. A. Krestov, Zh. Struki. Khim., 1962, 3, 137. '39 V. K. Abrosimov, Zh. Strukf. Khim., 1973, 14, 211; 1976,17, 838. '40 G. A. Krestov and V. K. Abrosimov, Zh. Strukt, Khim., 1964, 5, 510; Teor. Eksp. Khim., 1969, 5, 415;
G. A. Krestov and V. . A . Kobenin, ibid., 1973, 9, 680; V. K. Abrosimov, G . N. Makarov, and G. A. Krestov, Radiokhimiju, 1976, 18, 299; 1979, 21, 631; Zh. Sfrukt. Khim., 1980, 21, 120.
1 4 ' H. S . Frank and A. L. Robinson, J . Chem. Phys., 1940, 8, 933.
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132 Y. Marcus and A. Loewenschuss
Table 7 Transition temperatures, only below which ions are water-structure- breaking, according to Krestov and Abrosimov
Ion Li +
Na+ K + Rb+ cs +
c1- Br- I -
T,,, (H,O)/K (258)a 293 370
> 373 > 373
318 352
B 373
T*,IlS(D,O)/K (265>" 296 352 368
> 374 323 348
$374
Extrapolated value
considered that the bare aqueous ion retains the fraction K = 0.385 of its translational entropy in the gas (and all of whatever electronic and magnetic entropy it may have). Therefore
= AhydrS:bs - = AhydrS:bs f ( l - K)S?(g) (59)
For some of the ions it was found that AS:, was positive, these being the ions generally recognized as water-structure-breakers.
In later investigations by this group,'4o K was permitted to depend on the temperature, K = 0.363 + 0.002(t/"C), as found empirically for the inert gases, and as a result also ATI, was found to be temperature dependent. A table of the coefficients ai and b, of the quantity ASSlsh = ai + biT-', for the short-range component, was given for the alkali metal, alkaline earth metal, and halide ions, valid over the temperature range from 273.15 to 373.15K. It then transpired that this quantity, ASSIsh, may change its sign as the temperature was changed, and that the water-structure-breaking ions belonged to this category only below a certain transition temperature, characteristic for that ion. This transition temperature was sensitive to the isotopic nature of the water, whether it was H 2 0 or D20,I4 as shown in Table 7. At room temperature, it was found, the sodium ion is practically neither structure-making nor structure-breaking.
Abraham and c o - ~ o r k e r s ' ~ ~ used their multilayer hydration model (see Section 6) to evaluate contributions to the entropy of hydration, called by them AS: or AS& (differing by a small long-range electrostatic term). These involved both electrostatic and structural contributions, but the latter were emphasized in the interpretation. Positive values of these quantities were obtained for Rb', Csf, C1-, Br-, I-, and ClO;, and negative ones for Li+, Na+, K+, Ag+, and F- . The former group were the water-structure-breaking ions, the latter the structure-making ones, in good correlation with the Jones-Doles B-coefficientss4 and the Engel and Hertz B'-coefficients.'26 For these correlations, however, the original authors' assumption of the equality of the effects for K+ and Cl- was foresaken for a 'correspondence plot', in which B and B' values for cations and anions were forced to be located on
'42 M. H. Abraham, J. Liszi, and E. Papp, J . Chem. SOC., Faraduy Trans. I, 1982, 78, 197.
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Standard Entropies of Hydration of Ions 133
the same straight line, when plotted gainst S,". Furthermore, for the deri- vation of the S," values from the conventional ionic entropies of hydration, the unconventional highly negative values SP,,,(H+, aq) = - 34.7 JK-' mol-' was employed [for crystal ionic radii, used in equation (47), on the Pauling scale]. This may be the reason for obtaining K + among the structure-making ions, contrary to the generally accepted view concerning this ion (see Table 6).
Concluding Remarks. -The above discussion of the structrual effects on the entropy of ion hydration shows that none of the approaches suggested by the authors cited is inherently superior to the approach summarized in equations (51) to (57). The weaknesses of the latter have also been presented, but it is felt that its results, summarized in Table 6, justify its application. It may be noted, however, that for several ions no value of AstrucSP is given in Table 5, in addition to a few cases where values are given, but are not considered very reliable. These cases are discussed further below. On the whole, the cumula- tive errors residing in AstrUcS,O are as follows.
For monoatomic ions, S,O(g) can be considered as being accurate to < k 0.1 JK-lmol-' , for polyatomic ions to < k l.OJK-'mol-' (for exceptions see ref. 4). The reliability of SPconv(aq) varies from kO.8 to +_ 10 JK-'mol-' , being < k 3 J K-lmol-' for the bulk of the ions con- sidered. Hence AhydrSPconv should be reliable to < & 4 J K-' mol-', on the average. Transformation to the absolute values introduces a systematic error +_ 1.412, I J K-' mol-'. Together with the application of the Born equation this uncertainty rises to *31z,1, resulting in an average reliability of < k 3(2 + z?)li2 J K- ' mo1-' in AhydrSP*. The probable error in A,,,SP is 266n, J K-' mol-', the uncertainty in the logarithmic term of equation (55) being negligible, and an, is 0.355~~,~(r,~/nm)-~6(r,~/nrn). For an estimated error in the radius of 6rlC = &0.001 nm and for ions with rtc 6 0.06nm the error is 2.51~~1 J K- ' mol-' in AtnmS;, and it is smaller for larger ions. Thus the estimated overall reliability of AS,,,,SP is < 4( 1 + z?) ' '~ J K - ' mol-' for the bulk of the ions, i.e., < k 6 , < k 9, < 13, and < k 17 J K-' mol-' for uni-, di-, tri-, and tetravalent ions. For individual ions, of course, the estimated reliability can be completely different.
The cases of Li+, Be2+, and A13+ have already been mentioned, where apparently the calculation of n, according to equation (56) breaks down. For Li+, however, a minor adjustment (increase) to TIC makes A.,,,,,SP negative, and lets the ion be water-structure-making on this account, as it actually is. Similar in this respect are the cases of Sc3+, Cr3+, Co3+, and HP+, where minor adjustments to the radii confer the expected water-structure-making properties. For Ce4+ a major adjustment to the radius is necessary (from 0.080 to 0.100 nm), but this should not be ruled out, since the value quotedI3' has a question mark near it in the original Table. The larger value is also in line with the value for Th4+ and the lanthanide contraction.
14' Y. Marcus, unpublished results, 1982, presented in part at the IUPAC Conf. Chem. Thermodyn., London,
'4-1 Y . Marcus, J . Soh. Chem., 1983, 12, 271. 1982, abstr. 7-17; CJ Y. Marcus, 'Ion Solvation', Wiley, Chichester, 1985.
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134 Y. Marcus and A . Loewenschuss
Less readily accounted for are the apparent positive values of AstrucS," resulting for Pd2+, In3+, and Sn4+ (left out in Table 5) . It is contended that the fault may lie with the values of szon,(aq) used for these ions, that were obtained from the NBS Tables3 (see also comment on Pd2+ in Section 2). Two further ions, Sn2+ and S2-, are counted in Table 5 among the water- structure-breaking ones, on account of their positive Astruc s," values (two others, Ra2+ and Pb2+, are borderline case, in view of the general uncertainty limits discussed above). It is, again, suspected that the szo,,(aq) values employed3 are not sufficiently negative (see comment on S2- in Section 2). Inadequate correction for hydrolysis and activity coefficients may be respons- ible for these apparent failures with Sn2+, Sn4+, S2- (and perhaps also A13+). On the other hand, the negative values noted in Table 5 for A , , , , , ~ ~ for V3+, Ga3+, and T13+ though of the expected sign, seem not to reflect sufficiently the water-structure-making properties of these not-so-big trivalent cations.
The hydrogen ion presents a special case, since ni for it cannot be calculated from equation (56), hence neither can Atrim S," and A,,,,,SY. The hydrogen ion is generally considered to be water-structure-making, so that a maximal value of ni can be assigned to it, that still keeps it in this category. This maximal value is ni = 5.6 for the H+ ion, so that the tetrahydrate H(H20)4+, with four immobilized water molecules, would still comfortably be reckoned as a water- structure-making ion. If the ion H30+ is considered, then the radius of 0.13 nm assigned to it in Table 5 leads to ni = 2.7, or nearly three immobil- ized water molecules, leading again to H(H20)4+ that has a negative A,,,,,S," as expected. This tetrahydrate is recognized as the probably dominant (primarily) hydrated hydrogen ion.
Among the polyatomic ions there are several cases where the calculated A,,,,,SY value seems to be unreasonable, beside the great majority of the cases where it seems to agree with the known or expected structure-affecting properties. For O;, Ag(NH,);, RhCli- , and Fe(CN)i- the values are prob- ably either too positive or not sufficiently negative. Although their AstrucSY values are positive, according to Table 5 , they probably are not water- structure-breaking. For several more ions, however, the calculated A,,,,, s: is too negative or not sufficiently positive. These include the oxycations NOf, NO:, V 0 2 + , VO; , SbO+, ActO;, and ActO;+ (Act = U, Np, Pu, and Am), where much more negative values are noted than for smaller water-structure- making ions of similar charges. There exists the possibility that in spite of their positive charges these oxycations have strong propensities for hydrogen-bonding and the concomitant ordering of water molecules, con- siderably more than have the corresponding oxyanions (compare NO: with NO; !).
Some very large linear ions of relatively low charge: Hg;', S:-, Br; , and IF also have too negative values of AstrucS:, and at least for Br, and I; positive values are expected, in view of the positve values for Br- and I-. A re-examination of the S&(aq) values for these ions is warranted. (Inaccuracies in the assumed riC or ni values cannot account for the apparent anomalous AstrucSY for these univalent ions). The same, probably, can be said concerning the ions MnO; , TcO; , and ReO, , where not sufficiently positive
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Standard Entropies of Hydration of Ions 135
(for the former two) or a negative (for the latter one) values of A,,,, ,S~ are found.
The couple of ions Pd(NH),)i+ and Pt(NH,)i+ have negative value of AstrucSY differing by 273 J K-' mol-I, a difference that is unreasonable. Hydrogen-bonding of water molecules to the ammine groups should lead to a strong tendency towards water-structure-making, hence the value for Pd(NH,)i+ seems to be not sufficiently negative (see comment in Section 2). When the chlorocomplexes are compared, the values of RhCl2- (possibly also PdCli-) are too positive, that for IrC1:- not sufficiently negative, and that for ReCli- much too negative. However, the difference of - 60 J K-' mo1-' between the value for PdCli- on the one hand and PtCli- and IrC1;- on the other is also difficult to explain. There are no independent data (e.g., from viscosity measurements) to decide whether these ions are water-structure- making or -breaking, and, again, a re-examination of the $&(aq) values employed is in place.
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