STCDIES I N T H E SOLUBILITIES OF THE SOLUBLEELECTROLYTES
V. An Estimation of the Radii of Ions in Saturated Solutions BY ARTHUR F. SCOTT
For any given temperature the apparent molal volume p of a salt in a saturated solution containing one mol of salt is defined by the expression rp =
v - Nv,
(1)
where V is the volume of the solution; Tu’ is the number of mols of water in the solution; and v1 is the volume of one mol of pure water. I n a previous paper’ it was shown that for extremely soluble salts the following relationship between p and N is presumably valid: rp =
aN*
+b
(2)
The constants a and b are characteristic of each salt. The latter constant b is particularly important because it is the apparent molal volume of the salt when no water is present. Moreover, if the change in p with temperature is regarded as the consequence only of a change in the volume of the water present in the saturated solution, b may be looked upon as the volume of the ions of a salt in the saturated solution. I n the paper already referred to values of b were calculated for eight alkali salts and in magnitude they were found to be larger than the space occupied by the ions in a crystal lattice and smaller than the effective volumes of the ions in dilute solutions. I n the present article these considerations have been extended to permit an estimation of the radii of the individual ions. Such an estimation appears possible by making use of a new empirical relationship from which can be calculated the limiting volume V which contains the ions of volume b. Since the ratio b/t; is of the order of magnitude found for the crystalline state, we can, by assuming that the ions are arranged in a specific lattice, compute definite values for the radii of the ions. We shall first consider the following empirical relationship between V and N a t the same temperature: V K = aN B (3
+
I n order to test this relationship Fig. I has been constructed by plotting the experimental data for eight alkali salts. The essential data for this graph are collected in Table I. An examination of Fig. I shows that the plotted points for each salt, with very few exceptions, fall directly on a straight line. This method of plotting is, however, not very sensitive to experimental errors in the basic density and solubility measurements. Errors in density measurements will affect only the Vn term but, since they are much smaller than errors Scott and Durham: J. Phys. Chem., 34, 2035 (1930).
SOLUBILITIES OF THE SOLUBLE ELECTROLYTES
FI
1411
a
-
. . .
0 N
* h
-
mm
10m ,
h
10
roh
0.''
m - r . m
2
x
k-
N
roo
? ??
m mm w m
2 * . h .a w . 1. 0. h . m o m ha 10 mmN
c 0
ti N
N
m m
???
a-7m
z
3
rom
x
b
0
r o l n r o o 0 mm 1 0 N N m * w w w w r . a
1412
ARTHUR F. SCOTT
-36 -31
-32
-30
-%IO
d
-26 -24
-22
FIG.I The Variation of the Volume of a Saturated Solution with its Water Content N.
in solubilit'y measurements, they are probably negligible. Errors in the determination of solubility affect both terms, the value of S somewhat more than the corresponding value of V%. The few cases in which the plotted points do not fall on the line could all be attributed to errors in the solubility measurements. Xevertheless, until more data are available this means of testing the empirical relationship can not be regarded as decisive. Indirect evidence in support of the plausibility of equation (3) is t o be found by differentiating it with respect to S , as below:
If we take the Vn term to represent the average distance between the ions in a saturated solution, the expression means that the change in volume with N is directly proportional to the separation of the ions. This result is qualita-
1413
SOLUBILITIES O F THE SOLUBLE ELECTROLYTES
tively what would be expected on the assumptions of the electrostriction theory. If water (N) is added to a saturated solution in which the ions are relatively close, the added water will necessarily come into a region where the ionic forces are stronger and will consequently be more compressed than would be the case if the ions were farther apart. The increase in volume of the solution upon addition of water would be less in the first case than in the second. It is also noteworthy that the constant a turns out to be identical for certain salts, a fact which can be taken to mean that in saturated solutions of these salts the effect of the ions on the water molecules is essentially the same. If the salts of one group are considered, it is seen that they have in common one ion which is relatively stronger and which may be looked upon as the ion influencing the change in volume of the water molecules. I n other words we have an additional example of a possible dominant ion effect.’ Attention
TABLE I1 Values of , ! Iand Related Quantities NaBr
j3
-v b $
13.2 48.0 31.4 0.65
NaNOs
NsI
13.9
14.3 54.1 42.8 0.79
51.8 39.2 0.76
RbCl
CSCl
KCI
KBr
KI
15.9 63.4 40.4 0.64
16.3 65.8 48.4
16.9 69.5 34.3 0.49
17.3
17.9 75.7 54.7
0.74
72.0
41.6 0.58
0.72
should also be directed to the fact that a change in the solid phase is marked by a discontinuity in the linear relationship. For example, in Fig. I a dotted line is drawn to indicate the variation of V” with N for sodium bromide above the transition temperature. It is a t once apparent that the constants of this line (equation 3) are altogether different from those of the solid line, which correspond to the hydrated salt. For our purpose the constant j3 is particularly important and the values taken from Fig. I are given in Table 11. I n the same table are likewise given figures for V,which is equal to P3” The quantity V is the hypothetical, limib ing volume in which one mol of salt is contained when i t is in the same state as in a saturated solution but when no water is present. The corresponding values of b (equation z ) , which is the space occupied by the ions of an electrolyte under the above conditions, are also presented in Table I1 and are taken from the paper already mentioned. Finally, we shall find it convenient to introduce the quantity $ = b/V which is termed the space-filling quotient. The values of 4 given in Table I1 accordingly indicate the fraction of V which is occupied by the ions (b). It will be observed that, although the $ values vary widely among themselves, they are essentially greater than 0.5. This fact permits2 the preliminary deduction that the ions in volume V must be packed rather closely, in a manner similar to that of ions in a crystalline solid. For any mode of Scott: J. Phys. Chem., 33, 1000 (1929).
* R. Lorenz: “Rmmerfiillung und Ionenbeweglichkeit,” page 38.
1414
ARTHUR F. SCOTT
packing the quotient $ will vary with the ratio ri/rz of the radii (r, and rz) of the two ions involved. This variation, when the ions are assumed to be perfect spheres, is shown by the two solid lines drawn in Fig. 2 . The line marked C.S.= 8 is for the cesium chloride structure where the co-ordination number is eight; the line marked C.K. = 6 is for the simple rock-salt structure. The value of $ for a tetrahedral mode of packing (C.N. = 4) is indicated only for the one case where the radii ratio equals unity. Furthermore, the extreme limits of the drawn lines are established by the minimum permissible values of the radii ratios.’ The conditions which determine these minimum values
-01
-0.6
Y -0-5
KCI
-
w r z (15
05
~l
a8
~9
1.c
FIG.2 Variation of ‘Y with ratio of ionic radii.
are ( I ) any sphere (ion) of one sort must be in contact with the number of adjacent spheres (ions) of the second sort given by the co-ordination number and ( 2 ) the adjacent spheres must not intersect. In Fig. z are marked the $ values (Table 11) of the different salts. With the exception of the value of potassium chloride the indicated J. values can be plotted on the C.K. = 6 curve. On the other hand only two salts, potassium iodide and cesium chloride, have 4 values which can also be plotted on the curve drawn for the cesium chloride structure. In considering the significance to be attached to this graph it is important to keep in mind the two assumptions which are involved: first, the domain of the ions is spherical; and second, the oppositely charged ions are in contact. Since the values of b and V, obtained by extrapolation, are for indeterminate temperatures above IOO’C., the above two assumptions are only approximately valid. If, however, we continue to take b to be independent of temperature, a correction for the effect of temperature on V would tend to increase the magnitude of J.. Such a correction, for example, would bring the # value of potassium chloride nearer to the drawn curve; but the same correction applied to the $ value of sodium 1
Goldschmidt: “Crystal Structure and Chemical Constitution,” The Faraday Society,
(1929).
1415
SOLUBILITIES O F THE SOLUBLE ELECTROLYTES
iodide would make it too large. Although no estimation of the magnitude of this temperature correction can be made, one conclusion is worth noting: the effect of a temperature correction will be to decrease the radii ratio, that is, to increase the disparity between the radii of the constituent ions. With the values of b and V a t hand it is possible to compute the radii, rl and r2, of the ions constituting each salt by means of the following equations: (rl r2I3 = A V Xf ?r (rI3 rZ3)= b
+
+
Here S is Avogadro’s number; A is a constant dependent on the type of crystal structure. For rock-salt structure it has the value 0.938 X IO-$ and for cesium chloride structure its value is 1.02 X IO-$. The values of rl and r2 obtained by solving these equations, using the V and b values contained in
TABLE I11 Radii of Ions in the Limiting, Hypothetical State Salt Structure
rl x Io-8cm. r2XIo-8cm. rl/r2
NaBr NaNOa NaI NaCl NaCl XaC1
RbCl
1.07 2.43
2.j1
1.37 2.37
1.20
2.20
2.59
0.55
0.44
0.41
0.j8
0.46
1.21
1.04
NaC1
CsCl NaCl CsCl
KI NaCl CsCl
1.73
KBr NaCl I.j 9
1.28
1.85
2.40
2.31
2.69
2.47
0.72
0.69
0 . 4 8 0.75
Table 11, are given in Table 111. The radii ratios given in the bottom row are, of course, identical with those which would be obtained from the curves drawn in Fig. 2 by using the appropriate values of Only for the two possible cases, cesium chloride and potassium iodide, have the radii been calculated for the two types of crystal structure. No values for the radii of the ions in the case of potassium chloride are presented because the solution of the two equations yields imaginary values. This fact means merely, as has already been pointed out, that the oppositely charged ions are not in contact. It is of considerable interest to compare these values of the ionic radii with the values obtained in other ways. Although we have no means of knowing which of the two ions is the anion, we shall assume that the one (r2) with the larger radius is always the anion. The reasonableness of this assumption rests on the fact that practically every estimate of the relative magnitudes of the cations and anions under consideration indicates that the anion is larger than the cations. The values of rl and r2, interpreted according to the above assumption, are plotted in Fig. 3 : those values involving the assumption of a rock-salt structure are plotted as circles while those involving the assumption of cesium chloride structure are plotted as triangles. The salt from which each value is derived is also given. As a basis of comparison the ionic radii found by Goldschmidt’ (line A) and by Webb2 (line B) are shown in the figure. The values of the ionic radii portrayed by line A are those found for the ions in crystals; they are in excellent agreement with the values calculated
+.
page
Goldschmidt: “Crystal Structure and Chemical Constitution.” The Faraday Society, 282
(1929).
Webb: J. Am. Chem. SOC.,48, 2589 (1926).
1416
ARTKUR F. S C O W I
-2J
- 1
-2.6
-2.5
- 2-4 -2.3
-22 -2.1
-2.0 -19
-1.0
-U -1.6
-1.5 -1.4
-13
-u 1 . 1 40
1
II I I
I ‘I
I I
I
I
I
I I I
I
I
I I
I
I
;/...* %
I I I
I
I
I I I I
I
I
I
K’
Y
I I
I
‘
+I K i p
II
I/;I
I
I I
I I II
I
I I I
I
8; M
~
I
I
&a
I
N
a
K
R
I
b
G
I
I I
C
l
I I I
I I I II
I
I I I I
I I
h
I
NC
I I
I I
FIG.3 Calculated rahi of ions in L g s t r o m units.
from the theory of wave mechanics and with those deduced from the optical data of the crystalline salts. The ionic radii which determine line B are the effective radii of ions invery dilute solutions, calculated from the theory of electrostriction. The effective radius of an ion is understood to be “the radius of a sphere surrounding the center of an ion, inside which there are no water molecules.’’ We may therefore look upon these effective radii as the apparent radii when the ions are in the same state a s in very dilute solution but when no water is present. They would thus be comparable to the apparent radii obtained in this paper, provided that the effective volumes do not undergo a change with concentration. The plotted points in Fig. 3 appear to fall into two distinct groups: the anions have radii approximately the same as the effective radii calculated
SOLUBILITIES O F THE SOLUBLE ELECTROLYTES
1417
by Webb; the cations have dimensions very close to those found for the same ions in crystals. This generalization is strengthened by the observation made with respect to the temperature correction, namely, that such a correction would act to increase the ratios of the ionic radii. I n other words, a temperature correction would affect the calculated radii in such a way as to emphasize the division of the cations and anions intJotwo groups. Since we have postulated that b is the volume of the salt-ions in the saturated solut,ion, we must regard the calculated radii as approximately the effective radii of the ions in the saturated state. On the ground of this analysis it is allowable to suggest that concentration has no influence on the effective volume of anions whereas it does bring about a diminution of the effective volume of the cation. The foregoing interpretation is, of course, only tentative because of the limited data on which i t is based. -4further difficulty in the way of drawing a definite conclusion is the necessity of assuming a specific mode of packing of the ions in the limiting state. An illustration of this difficulty is the case of potassium iodide. Although the face-centered structure (C.K. = 6 ) , which is the structure of this salt in the crystalline state, gives results which agree excellently with the above interpret>ation,the body-centered structure (C.N. = 8), which is equally possible, gives a quite contrary result. In the latter caEe, as can be seen in Fig. 3, both ions have radii corresponding to those found by Kebb. I n this connection it is m-orth noting that the assumption of a body-centered structure for cesium chloride, which is the normal crystalline structure, yields reasonable figures for the ionic radii, whereas the assumption of the other structure yields abnormal, almost impossible figures. I n view of this evidence the conflicting results obtained in the case of potassium iodide on the assumption of an irregular but possible structure might conceivably be disregarded. rnfortunately, however, the evidence is by no means decisive and, as a matter of fact, only tends to emphasize the difficulty in question.
Summary In the present paper an empirical relationship between the volume (V) of a saturated solution and the solubility (X) of the saturating salt is discussed. From this relationship there is obtained by extrapolation the limiting volume v for the hypot,hetical state where no water is present. The apparent volume (b) occupied by the ions in this limiting state was previously found for eight alkali salts. The space-filling quotient b, V for these eight salts is of the same order of magnitude as that of crystalline solids. By assuming a lattice arrangement of the ions in this limiting state it is possible to calculate the radii of the individual ions. The apparent radii found in this way are, in general, larger than the corresponding radii in the crystalline state and smaller than in dilute solution. Finally, it is pointed out as a possible regularity that in the saturated state the effective volumes of the anions are the same as in dilute solutions while those of the cations are approximately the same as in the crystalline state. The Rice Institiiie Houslon, Texas.
