1304
M. J. BLANDAMER AND M. C. R. SYMONS
Val. 67
SIGNIFICASCE OF N E W VALUES FOR IONIC RADII T O SOLVATIOS PHESOllIENA I N AQUEOUS SOLUTION B Y ;\I. J. B L A N D A M E R AND 14.c. R. S Y M O N S Department of Chemistry, T h e University, Lezcester, England Received J a n u a r y 10, 1963 The behavior of ions with rare gas structure in solution is discussed in terms of radii derived from electron density maps of sodium chloride and those estimated indirectly. It is concluded that the former “experimental” values are preferable.
Introduction Although the term “ionic radius,” when applied to ions in solution, cannot be given a rigid definition,l nevertheless, many attempts have been made to relate the thermodynamic changes associated with the dissolution of an electrolyte in water to the radii of the positive and negative ions.* The major problem is to find a satisfactory method of dividing both the experimentally determined interionic distances in the crystal and the free energies of solvation in mater of the salt into separate contributions from cations and anions. Some workers, for example Bernal and F o ~ l e r and ,~ more recently, N o y e ~ have , ~ equated radii of ions in water to those suggested by Pauling6 for ions in crystals. Crystal Radii.-Radii of ions in crystals, as proposed by Pauling,b are derived in a semi-empirical fashion in order to obtain a set of values which are self consistent with observed interionic distances. The method is based on the postulate that the sizes of a pair of isoelectronic ions are inversely proportional to the effective nuclear charges operating on the peripheral electron shells. Radii so obtained are assumed to be independent of the counterion in the crystal and electron distribution maps obtained over thirty years agoe indicated that this assumption was valid for the fluoride ion in the series sodium fluoride, lithium fluoride, and c$cium fluoride, although the radius reported, L e . , 1.26 h., was smaller than that calculated by P a ~ l i n g . ~In contrast, however, the radius for the sodium ion was found to decrease from sodium chloride to sodium fluoride. On the basis of electron density maps obtained more recently by Witte and Wolfel’ for sodium chloride, Gourary and Adrian8 have arrived a t the new set of ionic radii for simple ions with rare gas structure shown in Table I. The maps demonstrate the ionic character of this salt and indicate only slight deformity of the ions from spherical symmetry. Further, the maps show that along the [loo] line joining the sodium and chloride ions, the electron density falls sharply to zero. The ionic radii were equated to the distance between the point of zero electron tensity and the center of either ion. This gave 1.17 -4.for T N a f and 1.64 8. for ~ c l - . Gourary and Adrian adopted these 1-alues and proceeded to calculate, from known interatomic distances, the radii of other ions. The values so obtained (1) K. H. Stern and E. S. Amis, Chem. Rev.,69, 1 (1959). (2) B. E. Conway a n d J. O’RI. Bockris, i n “RIodern Aspects of Electrochemistry,” J. O ’ R I . Bockris a n d B. E. Conway, E d . , Butterworths, London, 1954, p. 47. (3) J. D. Bernal and R. H. Fowler, J . C h e m P h y s , 1, 515 (1933). (4) R. XI. Noyes, J. Am. C h e m . SOC.,84, 513 (1962). ( 5 ) L. Pauling, “ T h e Nature of t h e Chemical Bond and t h e Structure of Molecules a n d Crystals,” 3rd. E d . , Cornell U u n ereity Press, Ithaca, h’. Y., 1960. (6) R. J. Havighuist, P h g s . Rev., 29, 1 (1927). (7) H. Witte and E. Wolfel, Z. phgszk. C h e m . (Trankfuit), BS,296 (1955). (81 B. S. Gourary and F. J. Adrian, SoZzd State Phys., 10, 127 (1960).
reproduce t’hese distances within an error of about 1% with one expection, lithium fluoride. I n this case, electron density mapsg do not shorn a position of zero electron deiisit’y between the ions but only a minimum and it is suggesteds that the values for the ionic radii of the lit’hium and fluoride ions, calculated on this basis, are too small because of the ability of the sinal1 cation to penetrate the large anion. In Table 11, a comparison is made of the interatomic distances calculated from the two sets of ionic radii together with those observed,1°and it would appear that the new radii give slight’ly better agreement. TABLE I COMPARISON O F L‘ALUES FOR IOXIC
FPayling6 radii, A. Gourary a n d Adrian8 radii, k.
C1-
Br-
I-
Li+
RADII
Ea+
K”
1.36
1.81 1.95 2.16
0.60 0.95 1.33
1.16
1 . 6 4 1.80 2.05
0.94
Rb”
Cs”
1.48 1.69
1.17 1 . 4 9 1 . 6 3 1.86
TABLEI1
Lk
COVP.4RISON
FC1BrI-
Obsd. Pauling Gourary Obsd. Pauling Gourary Obsd. Pauling Gourary Obsd. Pauling Gourary
O F OBSERVED”
AND CALCULATED INTERIONIC DISTAXCES,
and Adrian
and Adrian
and Sdrian
and ildrian
A.
Li+
Sa+
Kf
2.01 1.96 2.10 2.57 2.41 2.58 2.75 2.55 2.74 3.02 2.76 2.99
2.31 2.31 2.33 2.81 2.76 2.81 2.98 2.90 2.97 3.23 3.11 3.22
2.67 2.69 2.65 3.14 3.14 3.13 3.29 3.28 3.29 3.53 3.49 3.54
Rb+
Cs”
2.82 3.01 2.84 3 . 0 5 2.79 3.04 3.27 3.29 3.27 3.43 3.43 3.43 3.66 3.64 3.68
The new radii,* which for brevity are described as “experimental” do not differ greatly from those calculated by Pauliiig,6 but as shown in the following outline, the differences are sufficient to alter several correlations significantly. In Fig. 1, the “experimental” ionic radii and the radii of rare gas at’onisllare compared and it will be noted that the fluoride ion is now about equal in size to the sodium ion, rather than the potassium ion as suggested earlier.6 Ions in Solution.-In the following, various properties of ions in aqueous solution which have been related empirically or theoretically to ionic radii are discussed. (9) J. Krug, H. Witte, and E. Wolfel, Z. p h g s i k . C h e m . (Frankfurt), 4, 36 (1955). (10) K. K. Adam, “Physical Chemistry,” Clarendon Press, Oxford, 1956, p. 197. (11) These values are taken from ref. 4, Table 11, although other compilations, e.g., R. B. Heslop a n d P. L. Robinson, “Inorganic Chemistry,” Elsevier, London, 1960, p. 237, give widely different values for t h e radius of the helium atom. Hydride is omitted because its radius is x‘ery variable.
1303
SIGNIFICASCE OF NEWVALUES FOR IONIC RADIITO SOLVATION PHENOMENA
June, 1963
2.4
2.2
2.0
04 $
4
1.8
2 1.6
1.4
1.2
b
1.1)
‘i+
I
0.8
I
0.4
I
n
-1
I
0
0.8
Fig. 1.-Comparison of the radii of rare gas atoms with “experimental” radii of iso-electronic ions.
These haT;e been selected from a literature abounding in such correlations. Entropy of Solvation.-There have been many attempts to correlate partial molar entropies of solvation of ions in water with their radii and although the equations used differ in form, it seems12that most of these correlations are about equally effecthe in predicting experimental values. A common feature of such correlations is that “effective” ionic radii are proposed, with different increments added to the radii of cations and anions. Partial molar entropies of ions taken from the compilation of Powell and Latimer13 but adjusted to the scale proposed by Gurney14 have been plotted in Fig. 2 against 1 / ~ , 2 for both sets of radii. It seems that a single smooth curve is almost adequate for both cations and anions when “experimental” radii are used, so that it appears no longer necessary to differentiate between anion-water and cation-water interactions with respect to entropy considerations. Ionic Mobility.-That the limiting equivalent conductivities of ions with rare gas structure in water15 decrease with decrease in ionic radius is well known and has been discussed in classical terms as evidence for a corresponding increase in ion-solvent interaction through the series. However, the correlation for cations and anions collectively with “experimental” ionic radii is better than that with Pauling radii. This is shown in Table 111,where the ions are placed iii order of increasing size. (12) G X. Leiris a n d M. Randall, “Theimodynamlos,” 2nd. Ed., K. S. Pitzer and L. B r e u e r , Ed., ilIcGraw-Hill Book Co , London, 1961, p. 524. (13) R. E Pose11 a n d W. AI. Latimer, J . Chem. P h y s , 19, 1139 (1951). (14) R. W. Gurney, “Ionic Piooesses in Solution,” McGraw-Hi11 Book Co., London, 1953, p. 175. (1.5) R. A Roblnson and R . H Stokes, “Eleotiolyte Solutions,” 2nd. E d , R u t t e r w o ~ t l m ,London, 1959, p. 463
I
1
1.6
2.0
6
2.4
I
2.8
1/riz, .$.-a.
+1
Charge.
I
I
1.2
Fig. 2.-Correlation diagram between entropy of ~ o l v a t i o n , ~ ~ ~ ’ ~ in water and both Pauling, A, and “experimental,” 0,radii.
i‘
130
c
0.2
0.3
0.4
0.5 0.6 l/ri2,
0.7
0.8
0.9
1.0 1.1
1.1
K.
-2.
Fig. 3.-Correlation diagram between “experimental” ionic radii and free energies of solvation in water derived from the data for rubidium chloride, 0 , and those suggested by Latimer,, Pitzer, and Slansky,16 A.
TABLE I11 LIMITINGEQUIVALENT ~COSDUCTIVITIES OF IONS IS WATERAT 25”, C M . OHM-^ ~ EQIJI~.-’, ASD IONICRADII Pauling radii
(38.7) Lif N a + (50.1) (73.5) K’ (55.4) FR b f (77.8) cs+ (77.3) C1- (76.3) Br- (78.1) I(76.8)
Experimental radii
Li+
(38.7)
F(55.4) K a + (50.1) K + (73.5) Rbf C1Br-
cs+ I-
(77.8) (76.3) (78.1) (77.3) (76.8)
Free Energy of Solvation.-In deriving the contribut,ions of individual ions to the total free energy of solvation in water, the method introduced by Bernal and Fowler3 has been adopted. These authors divided the
GOOD,TODD, MESSERLY, LACIKA, DAWSON, SCOTT, ASD MCCULLOUGH
1306
heats of solvation on the basis of the similarity in Pauling radii for potassium and fluoride ions but, if the new values are adopted this method is no longer justified, and it is more reasonable to use rubidium and chloride ions as showii in Table I. In this manner, the total free energy of solvation, - 151.5 kcal./mole,16 for this salt has been divided into two equal contributions and the values for other ions obtained by difference through the chloride and rubidium series of salts. These values are plotted, in Fig. 3, against l / r I 2 Fince such a dependelice results if the main free energy contribution arises from the interaction between the ion and the contiguous water molecules.l7 Included on the graph are the free energies obtained by Latimer, Pitzer, and Slansky,IBwho fitted the free energies of solvation to the Born equation in terms of another set of effective ionic radii. The plot demonstrates that the simple procedure adopted here results in a dependence which does not differentiate between anion and cation. TABLE IV
FREE ENERGIES OF
SOLVATIOS I N TTSTER FOR ALKALI A S D
HALIDE Ioss Lit
Ka-
Kf
Rb+
Cs+
F-
C1-
Br-
I-
AG,,,,
- kcal. g. ion-1
122 9
97 6
81 3
75 8
67 0
103 3
75 8 69.2
61 8
Discussion When Pauliiig radii are used, it is found that anions invariably have greater free energies of solvation than cations of equal size and charge. In particular, dif(16) K. S. Latimer, I
