POLARIZABILITY OF ALKALIAND HALIDE IONS
3407
Polarizability of Alkali and Halide Ions, Especially Fluoride Ion
by Kasimir Fajans Department of Chemistry, University of Michigan, Ann Arbor, Michigan
48104 (Received October 87, 1969)
After it was recognized, in 1924,l that the Lorentz-Lorenz refractivity R (proportional to dipole polarizability a) of ions in crystals depends on the electric field of oppositely charged ions, num,erous attempts were made to derive values of these properties for free gaseous ions. Those proposed recently for alkali and halide ions by Wilson and Curtis (WC) differ considerably, especially for anions and particularly for F-, from previously derived ones. On the other hand, the polarizability CY of F- obtained by WC coincides with that calculated by Cohen, applying the wave mechanical coupled Hartree-Fock approximation. It is shown in the present article how a set of a’s for analogous ions can be tested for consistency, either by considering the deviations from additivity or by applying the inequalities method. Both ways lead to the conclusion that the new CY for F- (1.56 A8) is too large and that there is no reason to revise the value 0.95 used by the present author.
1. Introduction Wilson and Curtis (WC) acknowledge in a recently published paper2 our correspondence about an earlier version of their manuscript. Their work deals with the dependence of the polarizability CY of ions on the field of oppositely charged ions in alkali halide crystals. In a qualitative respect the view accepted by WC in the final version of their article agrees fully with the general principle which my coworkers and I applied to an extensive experimental material since 1924: the polarizability (initially the Lorentz-Lorenz molar refractivity RD)diminishes in the field of positive, increases in that of negative charges. Nevertheless, the same experimental polarizabilities of the crystalline alkali halides lead us to values for the free gaseous ions which differ in the following way. The ratio WC/BF is:s 0.79 for Na+; on the average 0.93 i 0.014 for K+, Rb+, Cs+; 1.24 0.02 for C1-, Br-, I-; for F- it has the large value 1.635. (See the individual values of a in Tables I V and We are especially concerned about the large discrepancy in the case of F- because the ( I ~ F value of BF was used recently as the basis of our ownsa,oand othersb investigations. The reason why various investigators arrive a t different a values for gaseous ions is that they attempt to satisfy different correlations. B F applied those derived by Fajans and Joos’ (FJ) which were confirmed by further theoretical considerations relating the polarizabilities of gaseous ions to the experimental values of the noble gases as well as by extensive data involving ionic compounds in crystalline, vapor, and dissolved state. The comparison of these experimental data with the chosen values for gaseous ions led always to reasonable deviations from additivity. Therefore in order to test the new set of CY values for free alkali and halide ions we shall apply to them both these methods.
*
2. Comparison of CY for Aqueous and Crystalline Alkali Halides First let us consider in Figure 1, as was done in Figure 1 of ref 1, the experimental differences between the alkali halides in aqueous solution and in the crystalline state. The resent figure applies to the polarizabilities CY in 3, the former one to RDin cm3. Numerous individual values, especially for the solutions, became more precise. Their sources in the present figure, in addition to those given in ref 2, are: for the solutions, ref 3 and 6; for the crystals, WC, Table I, column “Accept. ” The general appearance of the two figures is identical, except that in the present figure of the somewhat irregular’ values for the fluorides, namely Li -0.017, Na -0.082, K -0.096, Rb -0.085, Cs -0.051, the last two are not shown. For the interpretation of Figure 1 one has to take into account that each aaq- cycr for a given salt is the differential result of the following effects: tightening (decrease of a) of the anion in the crystal and of the
w
(1) K.Fajans and G. Joos, 2.Phys., 23, 1 (1924). (2) J. N. Wilson and R. M. Curtis, J . Phys. Chem., 74, 187 (1970). (3) The abbreviation BF refers to the paper with N. Bauer, J . Amer. Chem. Soc., 64, 3031 (1942),in which values of R m were derived. We apply the relation: R (oms) = 2.522 a (AS). Thevalues of a for free ions used here are those given by Wilson and Curtis in their Table I11 for Model 11. The F - value for Model I would give the still larger ratio 1.93. (4) During the editorial treatment of the present article a still different set of a values was published by A. J. Michael, J . Chem. Phys., 51, 5730 (1969). Contrary to WC, the values of AJM are larger than those of BF for the alkali ions (AJM/BF decreases from 1.60 for N a + to 1.24 for Cs+),smaller for the halide ions (AJM/BF increases from 0.80 for F- to 0.91 for I-). The assumed changes of a in the crystals do not show ri clear regularity. (5) (a) K. Fajans, Struet. Bonding, (Berlin), 3, 88 (1967); (b) R. A. Penneman, Inorg. Chem., 8, 1379 (1969); (0) K.Fajans, ibid., 8, 1553 (1969). (6) K.Fajans and R. Luhdemann, 2. Phys. Chem., ZQB,152 (1935). (7) The possible reason for the irregularity is an inaccuracy of the experimental R values of the hygroscopic crystalline RbF and CsF (see sections 3 and 6). The Journal of Physical Chemistry, Vol. 74,No.18,1970
3408
KASIMIR FAJANS values of R b F and CsF, which supports the assumption in ref 7. There are no evident inconsistencies in the ag values of BF. Table I : Change of the Sum of Anion in Solution and Crystal LiF
Figure 1. Polarizabilities, am,which alkali halides have in dilute aqueous solution minus those in the crystal. The somewhat irregular values for RbF and CsF are omitted for the sake of clearness of the figure. See text.
water molecules in the solution due to the field of the cation; loosening of the cation in the crystal and of the water molecules in the solution, due to the field of the anion; in the case of highly polarizable ions, like Iand perhaps Cs+, it is possible that anions are tightened and cations are loosened also in the field of the water dipoles. In spite of this rather complex situation there can be no doubt as to the meaning of the preponderance of positive ordinate values: the tightening of the anion by the cation in the crystals is the dominant effect. Taking into account that this effect can be expected to increase with the strength of the polarizing field of the cation and the polarizability of the anion, one understands the regular decrease of the positive ordinate values from Li + to R b + and from I - to C1-. The line for CsCl, CsBr, CsI is above that for the R b salts, obviously because of the different crystal structure of these three salts.8 Of special importanct? for our present discussion are in Figure 1 the negative, small values for the five fluorides. The negative sign of aq - cr can be due partly to a more pronounced tightening exerted by the cations on the water molecules in the solution than on F- in the crystal, and also to a more pronounced loosening of the cations than of water in the field of F-. In order to test these possibilities we shall compare the experimental aaqand a,? of the salts separately with the sums of the agof gaseous cation and anion according to B F as well as WC. 3. Comparison of Experimental a: with Those for Free Ions
Starting with the aq - g values in Table I part a, one sees that they are of the same order of magnitude as the experimental values in Figure 1. Those for LiF and NaF are negative, the other three are positive; this means that in the case of Li+ and Na+ the tightening effects predominate, in the case of K+, Rb+, Cs+ the loosening effects; even the gradation from LiF to CsF is regular. The latter is not the case for the cr - g The Journal of Physical Chemistry, Vol. 74, No. 18, 1970
aq cr
-g -g
-0.054 -0.037
aq cr
-g -g
-0.69 -0.675
NaF
01
(A8)for Cation and KF
a. ag of B F -0.046 0.055 0.036 0.151 b. a,of WC -0.61 -0.47 -0.53 -0.38
RbF
CsF
0.056 0.141
0.074 0.124
-0.43 -0.35
-0.41 -0.36
On the other hand, the WC values in Table Ib are all negative and the absolute values are considerably (on the average 74 times) larger than the absolute experimental aq - cr values in Figure 1. The latter appear to be a small difference of much larger effects involving the gaseous ions. This does not appear to be reasonable and the situation is very different when one considers, e.g., LiI, for which aq cr is 1.09. The latter difference compares as follows with those involving gaseous ions: g aq BF 0.25, WC 1.95; g - cr B F 1.34, WC 3.04. These values are of the same order of magnitude as 1.09, although those of WC are distinctly larger than the B F ones, because for a(I-, gas) the ratio WC/BF equals 1.23. Hence the great contrast for the fluorides seems to be caused by the unduly large CYT-for which WC/BF is 1.69.
-
-
4. The Method of Inequalities As the othet method for testing the F- value we shall use here that of "inequalities of quotients.'' It played a decisive role for the derivation of the R values for gaseous ions by FJ1 and has been explained since on many occasions in the research literatureea and textbooks.Ob The following example explains the main idea. Comparing the CY values of the isoelectronic F-, Ne, Na+, one has to take into account that the difference of 1+ between the nuclear charges is relatively larger between Ne and F- than between Na+ and Ne. One has therefore to expect for the quotients &' the inequality C Y F - / ( Y N ~> C Y N ~ / because ~ ~ ~ + 10/9(1.1111) > 11/10 (1.1000). This idea was extended in ref 9a to the quotients Q" and Q"'. Since in testing wave mechanical CY values in (8) K. Fajans, Z . Phys. Chem., 130, 724 (1927), where the first attempt was made to treat the deviations from additivity of RD quantitatively. See also in the WC paper ref 15-18 to investigations of Ruffa and others. The theoretical and quantitative aspects of these deviations will not be discussed in the present article. (9) (a) See especially K. Fajans, Z . Phys. Chem., 24B, 103, 114 (1934); (b) see, e.g., S. Glasstone, "Texbbook of Physical Chemistry," Van Nostrand, New York, N. Y., 1940, p 529; 1946, p 539.
3409
POLARIZABILITY OF ALKALIAND HALIDE IONS Table 11: Regularities of the Quotients for the Series 9 to 17 ~~
10
9
Q’ Q’, Qf”
12
11
17
16
-.
section 5 we shall proceed to still further quotients let us demonstrate here purely mathematically in which respect the inequalities are valid. Choosing as an example the series of the ions of neon type between the nuclear charges 9(F-) and 17(CI7+), the results are given in Table 11. It is seen that not only Q’, Q”, and Q‘” decrease from left to right but that each Q t ’ (Q”’) is smaller than both Q’ (Q”) above it. The irregularity indicated by --t in the series of the QIv values is caused by the insufficient accuracy due to the rounding off to four decimal places. In fact, the stated regularities apply to the further Q” the more decimals one chooses. But even with seven decimals QvIII is somewhat larger than the smaller Qvrl. Finally, with eight decimals the validity of the inequalities method becomes perfect for this series x = 9 to 17. 1.04183
1.0586
1.041726
1.05950
J QVIII
15
1.1000 1.0909 1.0833 1.0769 1.0714 1,0667 1.0625 1.0101 1.0283 1.0270 1.0260 1,0251 1.0244 1.0239 1.0218 1.0213 1.0210 1.039 1.037 1.035 1.035 1.033 1.031 1.032 1.032
QIV
QVII
14
13
1.1111
1.0597
1.05776
5. Testing of Wave Mechanical a Values Wilson and Curtis, comparing the (YF- value derived by them with wave mechanical calculation of Cohen, lo say: “It is therefore conceivable that even Cohen’s calculated value for a(F-), which agrees precisely with our extrapolated value (Model 11) is too low.” The inequalities method is very appropriate for testing of this assumption. Cohen used the coupled Hartree-Fock approximation in order to calculate a for F-, Ne, Na+, l!fg2+, and Al3+. His values agree closely with those of Lahiri and Mukherji” (LM) exce t that the latter include also for F- (1.40) is distinctly Si4+, and their value (in smaller than that of WC and Cohen’s (1.56). I n Table I11 the inequalities method is applied to the LM values. It is seen that all Q’ and Q” values are consistent with the inequalities rules and the Q’” and QIv diminish from left to right. However, the first Q”‘ and QIv as well as Qv are larger than the smaller of the values above them and the same is true for the second QIv.lla. In order to demonstrate how sensitive this method is, let us assume that the inconsistencies indicated by the three arrows at the left side of Table I11 are due mainly to the F-value 1.40 not being correct. One can achieve better consistency putting the smaller QIv equal t o 1.03
i3)
Table 111: Test“ of Lahiri-Mukherji’s
CY
in
Aa
I
F1.40
Q’ Q”
Ne 0.350
Ne‘ 0.140
Mgz’
Ala +
0.0607
0.0393
Si4
’
0.0241
4.00
2.500 2.008 1.773 1.631 1.600 1.245 1.1325 1.087
k/
Q”’
1.285
1.099
1.042
I(
I(
1.169
Q’V
1.055
k/ 1.108
QV
a The four arrows a t the bottom of the table indicate that the values involved increase in this direction instead of decreasing.
and &V equal to 1.02; there result as the first values on the left QIV 1.05 Q”’ 1.15 Q’’ 1.43 Q’ 3.58 F- 1.25 Accordingly the LM value 1.40 for F- and even more the WC value 1.56 appear to be too large. A more direct proof that the Hartree-Fock approximation is not exact enough for the derivation of the a’s for ten electron systems is that the L M 4 o h e n value for Ne (0.350) is 11% smaller than the experimental one (0.3953). This is not surprising because even for two electron systems the a’s calculated by Cohenlo do not agree with the precise values obtained by up to 96-term variational wave functions, as the following comparison shows. CHIZ Cohen’o
H-
He
30.50 13.8
0,2050 0.196
Li
’
0.02851 0.0280
The approximate values are smaller than the precise ones (in %) 1.8 for Li+, 4.4 for He, and 55 for H-. The pronounced discrepancy for H- was first shown by Schwartz. lSa The tremendous tightening effect ex(10) H.D.Cohen, J . Chem. Phys., 43, 3558 (1965); 45, 10 (1966). (11) J. Lahiri and A. Mukherji, Phys. Rev., 153, 386 (1967). (lla) NOTEADDEDIN PROOF.Considering the &‘ involving the a of Na’, Mgzf, Ala+in Table I11 and those of F-, Ne, Naf in Table V to be correct, the result is obtained that Qfa is approximately A table analogous t o Table 11, in which z = 9 to 14 equal are replaced by 98 to 14: shows a perfect validity of the inequalities rules down to Qv = 1.0212. (12) K. T. Chung and R. P. Hurst, J. Chem. Phys., 43, 35 (1965)
[CHI. (13! (a) C. Schwartz, Phys. Rev., 123, 1700 (1961); (b) K.Fajans, Chamoa, 13, 364 (1959).
The Journal of Physical Chembtry, Vol. 74, No. 18, 1070
KASIMIRFAJANS
3410 erted on H- by cations was emphasized earlier’3b using its approximate a. The precise value is reduced from 30.5 to 2.0 in crystalline KH, to 0.8 in the Hz (H- H+) m01ecule.l~
+
6. Application to a of Alkali and Halide Ions
The inequalities method can be applied to the WC and B F values in the following way. WC list in their Table I11 two sets of a for free alkali and halide ions. One was derived assuming that only the anions change their polarizabilities in the field of cations (Atode1 I), the other set attempts to take into account also the less pronounced variability of the cations (Model 11). For Li+ and Na+ the same theoretical values are listed in both sets, for the other ions the resulting a differ (in 89 between 0.02 for K + and 0.29 for Br-. We shall test here only the set I1 in which CYF-is smaller than in I. Of the values given for Cs+ (2.45-2.42) we choose the more favorable limit 2.42. The a’s of the noble gases in Tables IV and V are experimental.
Table IV:
Test of WC’s
CY
in
As
Q’“
F- 1.56 C1-4.41 Br-5.84 I- 8 . 9 1 Q’%
Q
3.95 2.687 2.350 2.217 Ne 1.479
Ne 0.3953
A 1.641 Kr 2.485 Xe4.019 A 1.328 I
1.121
Q”’c
a
.
Qlla
a
Test of BF’s
CY.
*
Q”
=
CY
in
As Q’
2.408 2.118 1.940 1.795 Ne 1.144
The meaning of
I
Q”’ = Q”(Ne)/Q”(av A, Kr, Xe).
Q”
F- 0.952 C1- 3.475 Br-4.821 I- 7.216
Rb+ 1.37 Cs+ 2 . 4 2 Xe 1.336
Av 1.319 f:0.016
Q‘ is the ratio between two neighboring
Table V :
N a + 0.148 K + 0.811
2.67 2.023 1.815 1.661 Kr 1.296
Ne0.3953.
A 1.641 Kr 2.485 Xe4.019 A 1.145
2.104 1.850 1.673 1.579 Kr 1.159
Na+0.1879 K + 0.887 Rb+1.485 Cs+ 2.545 Xe 1.137
Q’and Q” is the same as in Table IV.
The application of the inequalities method to a set of a’s corresponding to those in Table IV was discussed in ref 9a. It is to be expected that both kinds of the
The Journal of Physieal Chemistrv, Vol. 74, No. 18,1870
Q’ diminish distinctly, the Q” slightly from the Ne type to the Xe type. In Table IV the Q’ values fulfill this expectation but the Q’’ are somewhat irregular for the A, Kr, Xe types and their average (1.319) is considerably smaller than Q” (Ne) 1.479. This indicates that a for F- is relatively too large. To become consistent with the other values the ratio Q’(F-/Ke) should be 1.319 X 2.67 = 3.52 and a(F-) 1.39 instead of 1.56. In Table V the B F values are tested the same way. The comparison of the Q’’ for the Ne (1.144) and Xe (1.137) types shows the expected slight decrease, the only small irregularity is that the Kr value (1,159) is somewhat larger than both. An analogous situation was found in 1934O&when the method of inequalities was applied to RD. At that time R(Kr) appeared to be somewhat uncertain and it was shown that it would be sufficient to change the latter from 6.37 to 6.44 cm3 (1.1%) in order to avoid the irregularity. The remea~urement’~ of the refractivities of the heavy noble gases gave for RD of Kr 6.397 and of Xe 10.435 instead of the previous 10.42, but as Table V shows, these changes do not remove completely the irregularity of Q” for the a values. For the sake of comparison it may be said that in order to change Q” for the Kr type from 1.159 to the regular value 1.142 it would be sufficient to change a(Kr) from 2.485 to 2.504 A 8 (0.76%). This is out of the question but there are possibilities of small inaccuracies of the data involving some ions. For instance, it has been mentioned’ for other reasons that R of solid R b F may not be correct. If a reinvestigation would lead to a(Rb+) 1.463 instead of 1.485, Q’’ for the Kr type would become 1.142. The main result of the above discussion of Table V is that it does not contain any indication that a(F-) 0.95 is considerably too small. It may also be mentioned that this value nearly coincides with the average 0.88 zt 0.09 of values obtained by Hajj, Pauling, Ruffa, and Tessman-Kahn-Shockley, as listed in a recent investigation. l8 (14) The exceptional behavior of H - is due to the extreme ratio 2.000 of the negative:positive charges composing it. For the anions F - to I- this ratio decreases only between 1.111 and 1.019. Herice i t is not surprising that the inequalities regularities do not apply completely for the series H-, He, and Li+. For their above CH value Q” (20.69) is much larger than the smaller Q’ (7.190). More surprising is that even purely mathematically in a table analogous to Table I1 but for z values 1 to 4 the first Q”’ (1.1852) is distinctly larger than the second Q” (1.1250). These questions are beyond the main purpose of the present article. (15) D. Damkiihler, Z . Phys. Chem., 27B, 130 (1934). (16) F. Hajj, J.Chem. Phys., 44, 4618 (1966).
