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J. CHEM. SOC. FARADAY TRANS., 1991, 87(18), 2995-2999

2995

Thermodynamics of Solvation of Ions
Part 5.4ibbs Free Energy of Hydration at 298.15 Kt
Yizhak Marcus Department of Inorganic and Analytical Chemistry, The Hebrew University of Jerusalem, Jerusalem 9 1904, Israel

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The standard molar Gibbs free energies of hydration, Ah,vdGoI 109 (mainly inorganic) ions ranging in their of charges from -3 to +4 have been compiled and interpreted in terms of a model used previously for other thermodynamic quantities of hydration. The main contributions to AhydGo are the electrostatic effects, resulting in solvent immobilization, electrostriction, and dielectric saturation in a hydration shell of specified thickness, and further such effects on the water that surrounds this shell. Other effects contribute to AhydG0 to a minor extent only.

In a recent series of papers, the standard molar heat capacity,' entropy,2 and enthalpy3 of hydration of ions at room temperature (298.15 K) were examined. The single-ion thermodynamic properties were obtained from appropriate extrathermodynamic assumptions, and the validity of the tetraphenylarsonium tetraphenylborate (TATB) assumption for this purpose was examined These thermodynamic quantities of hydration were then interpreted on the basis of a common model.' This model had features that had already been suggested by others (e.g. Abraham and Liszi),6 but differed in some respects, and its applicability to all these quantities as well as to other ones (e.g. the partial molar volume) could be demonstrated. Its applicability to the standard molar Gibbs free energy of hydration was not presented in detail, however. Lists of standard molar Gibbs free energies of hydration of ions have been published.'g8 A more complete data base of these quantities has been assembled recently, and it is the purpose of this paper to present it and interpret it on the basis of the above-mentioned model.

Model
The model5 characterizes the ion by its charge, z, and its radius, r, and does not distinguish between cations (z > 0) and anions (z c 0) having the same radius and absolute values of the charge. The environment of the ion is divided into two regions: a hydration shell, in which the water is immobilized and electrostricted, and bulk water, that is, however, under the influence of the electric field of the ion. The thickness, Ar, of the hydration shell is specified as follows. The number n of water molecules in this shell is given by:
n = AlzJ/r (1)

The ion interacts with its environment in a manner that provides the following independent contributions to the thermodynamic quantities of hydration. A cavity in the water of radius r + Ar is formed, and the ion with its hydration shell is permitted to interact with the bulk water as if it were uncharged, i.e. by means of dispersion, dipole-induced dipole etc. forces. Then the charge is `turned on', and the electric field causes dielectric saturation in the hydration shell. The permittivity of the water is considered to follow a step function, having the value E' = ni (where n, is the refractive index at the D line) in the hydration shell and its bulk value I: beyond the distance r + Ar from the centre of the ion. Finally, effects of the ion on the structure of the water beyond those implicitly taken into account in the contributions just described are also recognized for the calculation of the standard thermodynamic functions of hydration or partial molar quantities of the aqueous ions. For the present discussion of the standard molar Gibbs free energy of hydration, these contributions take the following form. The `neutral term', representing the contribution of the interactions with the charge `turned off' is taken to equal the corresponding quantity for a suitable gaseous solute (e.g. a noble gas)? AG,,,JkJ mol-'
=

41 - 87[(r

+

Ar)/nm]

(3)

The Gibbs free energy of the electrostatic interactions in the hydration shell is given by: AG,,,
=

(N,,e2/8mo)z2(l - l/d)[Ar/r(r

+

Ar)]

(4)

and that for the interactions beyond it by: AG,,,
=

(Nav e2/8~~0)z2( l/~)/(r+ Ar) 1-

(5)

where A is the fitting parameter of the model, and is equal to 0.36 nm, as found for fitting AhydHodata.' Each of these n water molecules occupies a volume of nd3/6, where d = 0.276 nm is the diameter of a water molecule. Hence the volume of the hydration shell is: nnd3/6
=

For water at 298.15 K the sum of these terms is : AG,,, + ,/kJ mol=

-64.5z2[0.44(Ar/r)

+

0.987]/(r

+

Ar) (6)

(4n/3)[(r

+

Ar)3

-

r3]

(2)

where the right-hand side is the volume of the spherical shell of thickness Ar surrounding the ion of radius r. This thickness, Ar, is obtained by algebraic manipulation of eqn. (1) and
(2)-

with r and Ar expressed in nm.' The contribution of the effects of the ion on the structure of water to the Gibbs free energy of hydration beyond what is already specified by eqn. (3) and (6) is zero, since the structured and unstructured water around the ion are at equilibrium.' Hence the Gibbs free energy of interaction of the ion with its surroundings in the infinitely dilute solutions is just the sum of the contributions in eqn. (3) and (6):

t

Part 4:

Y.Marcus, J. Chem.

SOC.,Faraday Trans., 1987,83,2985.


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2996

J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

Table 1 The radius, r, width of hydration shell, Ar, number of water molecules in this shell, n, and the electrostatic contribution, AGc,1+2, calculated values [from eqn. (12)], AhvdG&, and experimental values, AhydG*, of the molar Gibbs energies of hydration of ions ion
~

r/nm
0.030 0.069 0.096 0.102 0.115 0.138 0.148 0.149 0.150 0.170 0.280 0.337 0.425
0.040 0.069 0.072 0.073 0.075 0.075 0.078 0.079 0.082 0.083 0.086 0.089 0.093 0.095 0.100 0.102 0.105 0.113 0.117 0.118 0.1 19 0.136 0.143

Ar/nm
0.300 0.172 0.125 0.116 0.097 0.074 0.065

n
12.0 5.2 3.8 3.5 3.1 2.6 2.4 2.4 2.4 2.1 1.3 1.1 0.8 18.0 10.4 10.0 9.9 9.6 9.6 9.2 9.1 8.8 8.7 8.4 8.1 7.7 7.6 7.2 7.1 6.9 6.4 6.2 6.1 6.1 5.3 5.0 20.4 17.4 17.7 17.4 16.9 16.6 16.1 15.4 14.4 13.7 12.6 12.4 12.3 12.3 12.1 12.0 12.0 11.9 11.7 11.5 11.4 11.3 11.1 11.0 10.8 10.7 10.7 10.6 10.4 10.3 20.3 20.0 18.0 15.5 14.8 14.4

H+ Li cu Na Ag K+ NHf Rb
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- 1058 - 558 -456
-440 -412 - 372 - 358 -356 - 355 - 328 - 221 - 186 - 149

- 1015

cs (CH,),N+
(C2HS)4N+

Tl+
+

0.064
0.063 0.049 0.014 0.008

Cr2 Mn2+ Pd2+ Ag2+ Sn2 Cd2+ Ca2 Hg2 Yb2+ Sr2 Eu2+ Pb2+ Sm2 Ba2+ Ra2

A13+ Cr3 Ga3+ Fe3+ Ti3+ Au3+ sc3 1n3+ Lu3 Yb3+ T13 Tm3+ Er3+ Y3+
HO~
+

v3

Dy3+ Tb3 Gd3+ Eu3+ Sm3 Pm3 Nd3+ Pr3 Ce3 Pu3 Bi3+ La3 Hf4+ Zr4 Ce4 Pu4 Th4+
+
+

u4

+

+

+

+

u3

+

+

+

+

co3
+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

v2

+

(C,H,),As Be2+ Ni2+ Mg2+ cu2 co2 Zn2 Fez+
+ +
+

0.004

-510 -400 - 385 - 350 - 305 -285 - 285 - 280 -245 - 75 0 125

- 1050 -475 - 525 - 365

+

+

+

+

+

-

- 295 - 285
-

430

275 -300 -250 - 160 0 50

0.322 0.233 0.227 0.224 0.220 0.220 0.213 0.21 1 0.205 0.203 0.197 0.180 0.183 0.180 0.171 0.168 0.163 0.150 0.145 0.143 0.138 0.118 0.109 0.324 0.296 0.299 0.296 0.29 1 0.288 0.282 0.275 0.262 0.253 0.237 0.235 0.233 0.233 0.23 1 0.228 0.228 0.226 0.223 0.220 0.218 0.216 0.214 0.212 0.209 0.207 0.207 0.205 0.201 0.203 0.306 0.303 0.283 0.253 0.245 0.239

- 3225

-2111 - 2049 - 2030 - 1992 - 1992 - 1940 - 1923 - 1876 - 1861 - 1819 - 1780 - 1731 - 1708 - 1656 - 1636 - 1608 - 1541 - 1511 - 1504 - 1497 - 1390 - 1352 5661 5007 5069 5006 -4886 -4829 -4722 -4569 - 4348 -4191 - 3954 - 3923 - 3893 - 3893 - 2864 - 3835 - 3835 - 3808 - 3780 - 3728 - 3702 - 3677 - 3653 -3629 - 3583 - 3561 - 3561 - 3539 - 3497 - 3476 - 7892 -7810 - 723 1 -6513 - 6333 - 6208

- 2005 - 1940 - 1920 - 1880 - 1880 - 1825 - 1805 - 1755
-

-3150

-

1740 1695 1735 1600 1575 1515 1495 1460 1385 1350 1345 1335 - 1210 -1160

-2395 - 1980 - 1830 - 2010 - 1915 - 1955 - 1840 - 1825 - 1850 - 1760 - 1910

-

- 1490 - 1755 - 1505

1865

-

-

-

1760 1510 1380 1385 1425 1375 1250 1250

0.053 0.062 0.06 1 0.062

0.065 0.067 0.070 0.075 0.079 0.086 0.087 0.088 0.088 0.089 0.090 0.090 0.09 1 0.092 0.094 0.095 0.096 0.097 0.098 0.100 0.101 0.101 0.102 0.104 0.105 0.07 1 0.072 0.080 0.093 0.097 0.100

0.064

-

- 5450
-

4965 -4830 - 4765 -4640 - 4580 -4680 - 4530 -4065 - 3895 - 3635 - 3600 - 3565 - 3565 - 3535 - 3500 - 3500 - 3470 - 3440 - 3380 - 3350 - 3325 - 3295 - 3270 -3215 - 3190 -3190 -3165 -3115 - 3090 - 7305 - 7215 - 6575 - 5755 - 5545 5395

- 4525 -4010 -4495 -4515 -4220 -4265 -4015 - 4420 - 3795 - 3980 -3515 - 3570 - 3970 -3515 - 3495 - 3450 - 3470 - 3425 - 3400 - 3375 - 3360 - 3325 - 3250 - 3280 - 3245 - 3200 - 3235 - 3480 - 3205 - 3145 - 6965 -6790 -6120 -6560 - 6360 -5815


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J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87 Table 1 (continued)
ion FOHHCO; CH,CO, HCO; NO;
BrO; CNNO; N, BrClO, H,PO, OCNSeH SHSCN1-

2997

r/nm

Ar/nm 0.079 0.079 0.059 0.055 0.050 0.044 0.043 0.043 0.038 0.038 0.037 0.035 0.035 0.033 0.033 0.032 0.032 0.03 1 0.029 0.026 0.023 0.02 1 0.02 1 0.02 1 0.0 19 0.017 0.004
0.076 0.070 0.059 0.043 0.039 0.038 0.032 0.020 0.018

n 2.7 2.7 2.3 2.2 2.1 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.8 1.8 1.8 1.8 1.8 1.7 1.7 1.6 1.6 1.5 1.5 1.5 1.4 1.4 0.9 4.0 3.9 3.6 3.1 3.O 3.0 2.8 2.3 2.3 4.5

AG,,,+,/kJmol-' 380 380 - 346 - 338 - 329 -317 -315 -315 - 303 - 303 - 302 - 299 - 297 - 293 - 293 - 290 - 288 - 285 - 279 - 272 -261 - 253 - 253 - 253 - 245 - 237 - 150
-

Ahyd

G,*,k/kJ mol
345 345 -310 - 300 - 290 - 275 - 270 - 270 - 260 - 260 - 255 - 250 - 250 - 245 - 245 - 240 - 240 - 235 - 230 - 220 - 205 - 195 - 195 - 195 - 180 - 170 15
-

-

'

AhydG*/kJ mol 465 430 - 335 - 365 - 395 - 300 - 400 - 340 - 330 - 295 - 330 - 295 -315 - 280 - 465 - 365 - 360 - 295 - 280 - 275 - 190 - 235 - 205 - 460 - 430 - 330 50
-

-

'

c1-

10;

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BF, MnO, ClO, BOY ClO, ReO, B(C,H

5

co;
S2-

1,

0.133 0.133 0.156 0.162 0.169 0.179 0.181 0.181 0.191 0.191 0.192 0.195 0.196 0.200 0.200 0.203 0.205 0.207 0.2 13 0.220 0.232 0.240 0.240 0.240 0.250 0.260 0.42 1 0.178 0.184 0.200 0.230 0.240 0.243 0.259 0.313 0.319 0.238

-

so; so:-

-

CrQSeOiSiFiPtCli PdCl 2 -

1195 1171 -1112 - 1010 - 979 - 970 - 923 - 787 - 744
-

1300 1280 1230 1145 - 1120 -1110 - 1075 -955 - 945
-

-

1315 1315 - 1295 - 1080 - 950 - 900 - 930 - 685 - 695 2765

Po;

-

0.054

-2153

2835

-

Data Base
Most of the values of the standard molar Gibbs free energy of hydration of the ions, Ahyd Go, at 298.15 K have been presented previously,' based on the conventional values (AhydG:onv[H+] = 0) in the NBS compilation." They are converted to the absolute values by means of the expression:
AhydG'

=

Ahyd

G:onv - 10562 kJ mOl-

'

(8)

corresponding to the choice AhydG'[H+] = - 1056 kJ mol-'. This choice, in turn, is based on the choices7 of the ValUeS AhydH'[H+] = - 1094 kJ mOl-' and AhydSo[H+] -131 J K-' mol-' or Sm[H+(aq)]= -22.2 J k-' mol- . The uncertainty in Ahyd G'[H+] is +6 kJ mol- and is the same as that of AhydH'[H+]. The latter value is consistent, though not identical, with the value preferred later,3 - 1103 f 7 kJ mol-', based on the TATB assumption. Conway' in his examination of the single-ion thermodynamic values of solvation chose the value AhydG'[H+] = - 1066 f 17 kJ mol-' on the basis of the data known at the time, consistent with our choice of - 1056 f 6 kJ mol-'. Therefore the conversion expression (8) is employed in the present study . Several further values are included in the data base, that are not available from the NBS compilation.'' These pertain to V2+, V3+ Ag2+, Ti3+, Cr3+, and Au3+, taken from Bartsch and Lagowski,' and several further values calculated from

7

'

'

Ahy,j

Go =

Ahyd

H'

-

T[S"(aq) - So@]

(9)

for (CH,),N+, (C2Hs)4N+, (C6H5),AS+, Sm2+,Yb2+, U4+, Pu4+, SeH- OCN-, BO,, Cloy, IO;, ReO,, (C6Hs),B-, HCO,, CH,CO;, HCO;, H,PO,, Cot-, SO:- SeOi-, CrOi-, SiFz-, PdCli-, PtCli-, and PO:-, with values of Ahyd H', Sm (aq), and S'(g) from previous publication^.^,^.' Altogether the data base included 109 ions, with charges, z, ranging from -3 to +4. For several further ions: C,H5NH:, (CH,),NHz, (C4H9)4N+, Mn3+, PO; and SbF;, although AhydH' is either S"(aq) or S'(g:l or both are unknown, so that Ahyd Go could not be calculated. The standard molar Gibbs free energy of hydration, AhydG', includes a term for the compression of the space available to the ion on its transfer from its gaseous to its aqueous standard states, that is foreign to the solvation (hydration) process proper. This process, that pertains directly to the interactions of the ion with its surroundings as specified by eqn. (7), is the transfer of the ion from a fixed point in the gas to a fixed point in the solution." Hence it is necessary to add RT ln[RT/I/,P,] = 7.93 kJ mol-' at 298.15 K to AhYdG',irrespective of the charge of the ion, to give Ahyd G*. In this expression V, = 0.001 m3 (1 dm3) is the standard volume of the aqueous solution and Po = 0.1 MPa is the relevant pressure. Admittedly, this correction is negligible in practice, but has to be applied in principle. The values of Ahyd G* are shown in Table 1, along with the values of rI3 and the derived values of Ar and n from eqn. (1) and (2). Since it is claimed that the model employed pertains to all the thermodynamic functions of hydration, the value of A in eqn. (1) obtained from the fitting of AhydHo data should


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2998

J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

8000 6000
4000

'i

B

1
9 D

on the right-hand side of eqn. (7), producing the values of
'hyd G,*,k

=

AGheut

+

AGell

+

2

+

AGunsym

(12)

%ih

0.00
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0.10

0.20

0.30

0.40

0.50

ionic radius/nm

Fig. 1 Values of the experimental -AhydG* (symbols) and calcuz lated -AGe,l+2 (lines) plotted against the radii r of ions: 0, = 1; .,z = -1; 0,z = 2; +,z = -2; A,z= 3; U,Z= 4

also be applicable to the fitting of the AhydGo. Eqn. (6) was used in order to obtain values of AG,,, + 2, and these are also shown in Table 1. The neutral term, AG,,,,, according to eqn. (3) is between 15 kJ mol-' for the smallest ion to 3 kJ mol-' for the largest. The fit of the electrostatic term (continuous lines) to the experimental values of AhydG* (symbols) is shown graphically in Fig. 1.

Discussion
Examination of Fig. 1 and of Table 1 shows that the model fits the experimental results remarkably well, considering its simplicity and the wide range of ionic charges, -3 to 4, and sizes, 0.03W.425 nm, included. However, a detailed look at the entries in Table 1 shows some discrepancies. One source of discrepancy is the linear decrease of AG,,,, specified in eqn. (3), according to Abraham and Liszi.6 Inspection of their paper, however, shows that this trend is valid only for the noble gases, but that for hydrocarbons and similar large and globular molecules the dependence of AhydG* of the non-electrolytes on their radii has an upward swing. These data conform rather to:
AGh,,JkJ mol-'
=

41 - 87(r/nm)

+

1200(r/nm)2 (10)

This leads to non-negative values of AhydG* for the largest ions, as is indeed observed. For such ions the amount of work done to create the cavity in the water in which the ion finds itself is larger than the Gibbs free energy released when the large ion interacts with the water through its charge and through dispersion and induced dipole interactions. Note that if for the ions the bare ionic radius r is used in eqn. (10) instead of r Ar specified in eqn. (3), this will have a negligible effect for the largest ions. However, this expedient is necessary for ordinary ions (with r < 0.25 nm), since otherwise (use of r + Ar) an over-correction results. This makes this correction empirical, since the cavity should have the size specified by r + Ar rather than by r. Another systematic discrepancy can be noted when the sum of the contributions from eqn. (6) and (10) is compared with the experimental values, mainly for the multi-charged ions. The cations tend to have more negative and the anions more positive calculated values. This tendency could have its origin from the fact that the water molecules in the hydration shell between r and r + Ar are oriented differently towards cations and anions. This fact has not been taken into account5 in the model, and is also at variance with the more simple-minded version of the TATB assumption. Empirical cognizance of this unsymmetric charge effect can be made by inclusion of the term:

+

AGunsym 120(r/nm)z3 =

(11)

shown in Table 1. The odd power of z in eqn. (11) causes the charge-unsymmetry correction to be negative for anions and positive for cations, and to be more pronounced for the more highly charged ions (and also the larger ones). This correction removes most of the systematic discrepancies, with fits generally to considerably better than +1001z1 kJ mol-'. This leaves only more or less non-systematic cases to be explained on an ad hoc basis. These cases include the smallest cations: Be2+,A13+,Cr3+, Ti3+ and Fe3+, that show large negative deviations of the calculated from the experimental values, as well as some larger ions: Cu+, Hg2+, T13+, Bi3+, Pu4+, U4+, IO,, H2P0,, BO,, ClO,, ReO,, PtCli- and PdCli-. A plausible explanation in the case of the smallest cations is that some of the many water molecules (n > 16) are sufficiently remote from the ion because of the crowding of the first shell, so that a larger E' than nk applies to them. Hence a less negative value of AG,,' + should have been used. Relatively large positive deviations of the calculated from the experimental values are shown by the four cations, Cu+, Hg2+, T13+ and Bi3+,that belong to the very soft group of cations (even Ag+ that belongs to this group shows a positive deviation, though < 100 kJ mol- '). However, no good reason is apparent why water, that is a hard ligand, would cause more negative values of AhydG* than the model allows. Reasons for the remaining outlying values could be inaccuracies in the ionic radius or the AhydG* employed. The former inaccuracy may apply in particular to the anions, where so-called thermochemical radii have been used. A case of good conformity of the calculated with the experimental value should be pointed out, since it is artificial, in the sense that the radius r was arbitrarily assigned. This is the case of the hydrogen ion, H+. Fine-tuning of r might have caused even better agreement, but was considered unnecessary. Non-conformity of the calculated values for the tetraphenylarsonium and tetraphenylborate ions, 125 and 15 kJ mol-l, with the TATB model, that calls in its simplest version to equal values (here Ahy, G* = 50 kJ mol- ') requires some comment. These values of 50 kJ mol-' were obtained by eqn. (9) from AhydHo = -47 kJ mOl-' and AhydSo = -302 J K-' mol-'.'6 The former of these appears to be better established than the latter, that is based on the application of the TATB assumption to Sm(MPh4).Whereas the standard molar entropy of a solid compound of Ph,B- is known, hence also the corresponding entropy of solution and Sm for this compound, this is not the case for any compound of Ph,As+ (or Ph,P+). Therefore this assumption cannot be checked with respect to its conformity with the established value of S"(H+) = -22.2 J K-' mol-I on which eqn. (8) is based. An uncertainty therefore is connected with the 'experimental' AhydG* values for the tetraphenylarsonium and tetraphenylborate ions. This uncertainty is not expected to be sufficiently large, however, to explain the difference of 110 kJ mol-' between the calculated values, that arises directly from the factor r in the empirical eqn. (ll), that appears to require it for ions with r < 0.3 nm. No manipulation of the simple form of this expression is warranted, however, for the special cases of the tetraphenyl ions. On the whole, however, the model permits remarkably good predictions of AhydG* of the ions, given the empirical adjustments in eqn. (10) and (11) relative to the simpler model used previously for fits of AhydH* etc. Therefore some physical meaning may be ascribed to the parameters of the model, in particular to n and Ar. 'Hydration numbers' are defined


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J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87
200
0.45

2999

160
r

0
I

/
0

0.40

N

E 120 E
80

0.35

5
-i

0.30

A

2 +

5

E

however, that the conductivities of cations and anions fall on the same curve and that the equivalent conductivities of all the ions would nearly do so. The non-coincidence of the extrema for the two variables may be due to the fact that the anions are generally larger than the cations, and that structure-making by the larger ions would affect the conductivity but not AhydG*.

References
1 M. H. Abraham and Y. Marcus, J. Chem. SOC., Faraday Trans.

40

0.25

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0 0.00

0.10

0.20

0.30

0.40

0.20 0.50

r/nm

2 3 4 5 6 7 8 9

Fig. 2 The molar conductivities and `hydrated radii' of ions plotted against their radii: circles, 1'; triangles, r + Ar; empty, z = f 1 ; filled, z = k 2

operationally only with respect to the method by means of which they are determined. The values of n resulting from the model can be considered as useful `hydration numbers', combining for multi-charged ions the values for the nearest neighbours with those of the next-nearest neighbours, as determined by X-ray diffraction and similar methods." The physical significance of Ar, or rather r + Ar, can be seen in Fig. 2, where the molar conductivity of the uni- and di-valent ions of both kinds of sign and their values of r + Ar are plotted against their r values. The former dependent variable shows a maximum approximately where the latter one has a minimum, implying the well known fact that it is the hydrated ion that moves in the electric field. It is significant,

10 11 12 13 14 15 16 17

I, 1986, 82, 3255. Y. Marcus, J. Chem. SOC., Faraday Trans. I, 1986,82,233. Y. Marcus, J. Chem. SOC.,Faraday, Trans. I, 1987,83,339. Y. Marcus, J. Chem. SOC.,Faraday Trans. I, 1987,83,2985. Y. Marcus, Pure AppI. Chem., 1987,59, 1093. M. H. Abraham and J. Liszi, J. Chem. SOC., Faraday Trans. 1, 1978,74,1604; 2858. Y. Marcus, Ion Soloation, Wiley, Chichester, 1985, ch. 5. H. L. Friedman and C. V. Krishnan, in Water: A Comprehensive Treatise, ed. F. Franks, Plenum Press, New York, 1974, vol. 3, ch. 1, pp. 54ff. A. Ben-Naim, Water and Aqueous Solutions, Plenum Press, New York, 1974, ch. 7. D. D. Wagman, W. H. Evans, V. B. Parker, R. H. Schumm, I. Halow, S. M. Bailey, K. L. Churney and R. L. Nuttall, J. Phys. Chem. Ref: Data, 1982, 11, Suppl. No. 2. B. E. Conway, J. Solution Chem., 1978,7, 721. S. G. Bartsch and J. J. Lagowski, J. Solution Chem., 1987, 16, 583. Y. Marcus and A. Loelwenschuss, Annu. Rep. C, 1984, 1985,81. Y. Nagano, H. Mizumo, M. Sakiyama, T. Fujiwara and Y. Kondo, J. Phys. Chem., 1991,95,2536. A. Ben-Naim and Y. Marcus, J. Chem. Phys., 1984,81, 2016. Y. Marcus and A. Loewenschuss, J. Chem. SOC., Faraday Trans. I, 1986, 82, 993. Y. Marcus, Chem. Rev., 1988,88, 1475.

Paper 1/02148F; Received 8th May, 1991