ii RELATIOKSHIP BETWEEN ATOMIC NUMBERS AND T H E PROPERTIES OF IONS I K THE CRYSTAL LATTICE1 BY ARTHUR F. SCOTT
The various relationships between the properties of ions and their atomic numbers have formed the subjects of a number of papers published during the last few years. The highly suggestive papers of Bragg2, Davey3, and Grimm4 in particular have pointed out the relationship between the radii of ions and their atomic numbers, a relationship which the present communication purposes to establish in a somewhat more general, quantitative form. Of the factors that determine the distance between ions, the point centers in a lattice, the two primary ones appear to be the dimensions of the ions proper and their external electrostatic fields which control the equilibrium position assumed by ions in a crystal. Since both of these are characteristic properties of an ion, they must be intimately connected with the structure of the atom. It is one of the canons of modern atomic theory that the internal structure of atoms up to the periphery is regulated exclusively by the positive nuclear charge, the atomic number. Positive ions, therefore, formed by the loss of peripheral electrons may be considered to have dimensions dependent on the atomic number. For the same reasons we may believe the residual, external field of an atom to be similarly related to the atomic number. But with an ion, the external action is also dependent on the number of electrons lost by the original, neutral atom; and, consequently, since periodicity is a function of the number of electrons in the outer shell, we would expect the external field to exhibit distinct periodic changes, and at the same time to undergo a regular variation with the atomic number. If our premise is correct, that the lattice distance of a crystal depends on the dimensions and external fields of the constituent ions, we may then, on the basis of the above reasoning, make the general statement that the controlling factors in determining the distance between a positive ion and a negative ion (anion or electron) are the atomic numbers of the two ions. A confirmatory illustration of this statement, is found in the familiar fact that the atomic volumes of isotopes are identical. Because of our fragmentary knowledge of the dimensions and external actions of ions, it would be difficult to establish what function the distance is of the atomic numbers. Nevertheless, an empirical rule which does express this relationship to an apparently quantitative degree is contained in the following general equation : d o = a In (Z,,-Z*)+b. Contribution from Department of Chemistry of Reed Collegr.
* Bragg: Phil. Mag., (6) 40, 169 (1920).
3 Davey: Phys. Rev., 23, 318 (1924). 4Grimm: Z. physik. Chem., 98, 353 (1921.)
(1)
BTOMIC WUMBERS AND PROPERTIES O F IONS
305
where do is the shortest distance between ions; Z,-2, represents the absolute value of the difference of the atomic numbers of the cation and anion; and a and b are constants. We thus have a linear relationship between the lattice distance and a function of the atomic numbers. However, in order to avoid any misunderstanding, it must be understood that there are sound theoretical grounds for believing that the above variation with the atomic numbers in any series mill not always be regular but will show a marked discontinuity in the majority of cases. The reason for this ciiscontinuity is to be found in Bohr’s suggestion that in the elements following argon the electrons instead of always going into the outer shell commence to fill up the vacant places in the inner shell. Such a change in the sequence wherein the orbits are filled up must also be related to the magnitude of the atomic number. Nevertheless the elements that precede and follow argon differ so in performance, that it appears that the radii and external field of the ion cannot vary in the same manner in both cases:-even though they do vary regularly with the atomic number. Under these circumstances we would expect a discontinuity to appear in any group a t the element with an ion structure similar to argon; and in the periods following argon we would also expect irregularities to appear corresponding to divergences from the normal rule on the part of the electrons. For the negative ions, such as the halogens, essentially the same principles would be true. By far the most diverse group of crystalline substances to test Equation I with are the metallic elements. It is relatively simple to picture the lattice structure of one of the alkali metals if we consider it to be essentially similar to that of sodium chloride with the chlorine ion replaced by an electron. It is not so easy, however, to devise a satisfactory representation for more complicated cases, such as silicon, where we have four valence electrons to place. For want of a more definite hypothesis it will be necessary to make two assumptions in order that all the metals can be treated in the same manner. First, we will postulzte that the negative charge of all the electrons is localized in one unit. For example, in the case of silicon with its four valence electrons, we will think not of four electrons but of one unit with four negative charges which we may term an electron-ion. Kotv it is patent that with this conception all the elements could be treated in the same manner as the alkali metals, were it not for the fact that the positive ions of the different elements have been shown to form various types of lattices. We shall therefore have to make a second assumption: We will imagine that an electron-ion is located between every pair of positive ions and that this electron-ion determines the distance between the positive ions (the equilibrium status) just as in the crystal of sodium chloride, the distance between the sodium ions has been shown to depend on the chlorine ion. One other aspect of the problem must be mentioned here. I n some cases, as in the hexagonal lattice the distance between two positive ions is not the same for the different neighboring ions. There is, however, a minimum distance to which they can approach and it is this minimum distance which is implied in the second assumption above. Such a portrayal is not inconsistent with the general properties of
ARTHUR F. SCOTT
306
TABLEI Nearest Approach of Ions in Metals Element
Z Z+e
In @+e) 0.693 I ,386 I ,792 2.303 2 ,485 2.639 2.773 2.890 2.890' 2.996 3.091 3.258 3,332 3.466 3.526 3.526 3.584 3.584 3.638 3.689 3.784 3.8j1 3.932 3.871 3.951 3.989 3.989 4.025 4.060 4.094 4.111 4.127 4 . I43 4.174 4.35j 4.382 4.431 4.394 4.431 4,454 4.543
Type of Lattice Authority Known or assumed Helium 2 2 4.13 Rock-salt Calculated Lithium 3.03 Body-centered cubic Hull' 3 4 Beryllium 2.28 Hexagonal McKeehanZ 4 6 Carbon 1.50 Hexagonal Debye3 6 IO Sodium 4.00 Body-centered cubic Hull' 11 1 2 Magnesium 3.22 Hexagonal Hull' 12 14 AIu min um 2.86 Face-centered cubic Hulll 13 16 Silicon 14 18 2.35 Tetrahedral cubic Debye4 Hull1 18 18 Argon 5.26 Rock-salt Calculated Potassium 19 20 4.48 Body-centered cubic McKeehan2 Calcium 20 22 Face-centered cubic Hull5 3.93 Titanium 2.90 Hexagonal Hull6 22 26 Vanadium 2.64 Body-centered cubic Hull' 23 28 Manganese Body-centered cubic Calculated 2.53 25 32 Iron 2.48 Body-centered cubic Hull1 26 34 Gallium Face-centered cubic Calculated 3.03 31 34 Germanium Tetrahedral cubic Kolkemeyers 2.43 32 36 Krypton 5.60 Rock-salt Calculated 36 36 Rubidium Calculated 4.93 Body-centered cubic 37 38 Strontium Face-centered cubic Calculated 4.33 38 40 Zirconium 3.18 Hexagonal Hull6 40 44 Molybdenum Hull9 2.72 Body-centered cubic 42 48 Ruthenium 43 51 2.64 Hexagonal ~u119 Silver 2.88 Face-centered cubic VegardlO 47 48 Indium 3.24 Tetragonal face-centered Hull9 49 52 Tin Tetragonal cubic BijP 2.80 50 54 Xenon 5.84 Rock-salt Calculated 54 54 Cesium 5.34 Body-centered cubic Calculated 5 5 56 Barium j 6 58 4 ,j z Face-centered cubic Calculated Lanthanum 3,j5 Face-centered cubic Calculated 5j 60 Cerium 58 61 3.64 Face-centered cubic Hull6 Praseodymium 59 62 3,71 Face-centered cubic Calculated Neodymium 60 63 3.65 Face-centered cubic Calculated Samarium 62 65 3.56 Face-centered cubic Calculated Tantalum 2.83 Body-centered cubic Hull9 73 78 Tungsten 2.73 Body-centered cubic Debye12, Hull9 74 80 Osmium Hexagonal Hull6 2.66 76 84 Gold 2.88 Face-centered cubic Vegard13 79 81 Thallium 81 84 3.38 Face-centered cubic Becker and Ebert14 Lead 82 86 3.48 Face-centered cubic Vegard13 Thorium 3.54 Face-centered cubic Bohlin16 Hull6 90 94 Uranium 2.99 Body-centered cubic CaIculated. 92 98 4 . $35 1 Hull: Phys. Rev., 10, 661-(1-917). 2 McKeehan: Proc. Nat. Acad. Sei., 8,270 (1922.) 3 Debye: Physik. Z., 18, 291 (1917). Debye: Physik. Z., 17, 277 (1916). 6HuIl: Phys. Rev., 17, 12 (1921). 6Hull: Phys. Rev., 18, 88 (1921). 7 Hull: Jour. Franklin Inst., 193, 189 (1922). Kolkmeyer: Proc. Roy. Acad. Sci., Amsterdam, 21, 405 (1919). 9Hull: Phys. Rev., 17, 571 (1921). 'OVegai-d: Phil. Mag., (6) 31, 86 (1916). Bijl: Proc. Roy. Acad. Sei., Amsterdam, 21, 501 (1919). l2 Debye: Physik. Z., 18, 483 (1917). l 3 Vegard: Phil. Mag., (6) 32 65 (1916). 14Beckerand Ebert: Z. Phykik, 16, 16.5 (1923). 16Bohlin: Ann. Physik., 61, 421 (1920). 2
du
.
S T O M I C NUMBERS AND PROPERTIES O F IONS
307
metals. For instance, if we conceive of all the electrons lumped together in electron-ions (which contain the maximum number of valence electrons) and if we think of the nearest approach of the positive ions as being determined by these electron-ions, then we can imagine electrons either in the form of these maximum units or singly, swarming throughout the lattice of positive ions without disarranging the structure permanently, the situation which obtains during the process of electrical conduction. The above hypothesis has been developed to make the application of Equation I to the metals more understandable. Therefore, to evaluate the term Z,-Z, which is equal to the difference in the atomic numbers or the positive charges of the ions concerned, it is necessary to ascribe a negative value to the Z of the electron-ion which will be equal to the number or valence electrons. I n short, Z = -e where e represents the number of valence electrons or the maximum valence of the element. Since e never exceeds the atomic number of an element, we can hereafter write Equation I in the form:
do= a In (Z+e)
+ b.
(2)
From the foregoing deduction it will be seen that do is equal to one half of the distance which is commonly termed “the nearest approach” of ions. To avoid subsequent ambiguity this distinction will be maintained although the distances employed in the case of the metals will always be the “nearest approach” or 2 do. Figure I depicts the results obtained when the distance between the ions, 2do, is plotted against the natural logarithm of the term, (Z+e). The source of the values for do is given in Table I which also contains the other essential data for the construction of Figure I . It seemed desirable to supplement the meagre data from X-ray analysis and therefore a number of values for the do were computed in the usual manner from the atomic volumes of the elements. I n all of these cases the value for the density employed in the calculations is taken from Landolt-Bornstein’s Tabellen; and the type of lattice assumed is in each particular instance determined by the types of lattices known for the neighboring elements. The results of these two modes of determining lattice distances are usually in satisfactory agreement for metals and consequently they may be considered to be comparable. Only when an element has no definite crystal structure under normal conditions is any serious discrepancy likely to occur. Potassium is such a case, for McKeehan has found that this element has a crystal structure only at low temperatures, and he made a measurement of the lattice distance a t about - 15oOC. With the atomic volume derived from this measurement it is possible to use the cubical coefficient of expansion of sodium as a rough approximation in order t o get the value for do a t a comparable temperature. This calculation has been carried out and since the values for three of the alkali metals are therefore necessarily obtained from calculations, those for lithium and sodium have also been calculated with the unusually precise density data so that this series can be on a more comparable basis. The solid inert elements were assumed to have a structure analogous to sodium chloride inasmuch as the salts formed by the
308
AHTHCR F. SCOTT
ions of adjacent elements such as barium sulfide all exhibit the same tendency to form the simple cubic lattice. This assumption was made in spite of the preliminary notice published by Simon and von Simsonl that solidified argon forms face-centered cubic crystals. The densities at absolute zero of the solid inert elements used in the calculations are those computed by Hertz2 according to the method of Lorentea which involves the density of the substance a t its boiling point, melting point, and critical temperature.
-I
1
NA TJRAL L OGAR/THM O f frt e,)
7
2
4
5 I
FIG. I
The Periodic Variation of the Ion Radii of the Elements.
Figure I displays a new means of portraying the periodic variation of the radii of the ions. The horizontal lines are drawn through the points corresponding to the elements of each group; and the vertical lines place the elements of the different periods. The slight divergences exhibited by cesium, beryllium, and magnesium, must be attributed to the suspected inaccuracy of the data. The general characteristics of the two series of lines are quite apparent. Up to and including the fourth group we find all of the elements varying regularly with ln(Z+e) with the predicted discontinuity a t the ion with the strucSimon and von Simson: Natuiwissenschaften, 11, 1015 (1923). 2Hertz: Z. anorg. Chem. 105, 171 (1919). a Lorenta: Z. anorg. Chem., 94, 240 (1916).
ATOBIIC XCMBERS AXD PROPERTIES Oh’ IONS
309
ture of argon appearing at potassium, calcium, scandium, and zirconium, respectively. In the groups beyond the fourth the first two elements are wanting due to their non-metallic nature and the remaining elements exhibit a distinctly regular, although different variation. Concerning the vertical lines it is only necessary to ncte that the heavy black line marked T indicates the location of the points for praseodymium, neodymium, and samarium, which fall accurately and so closely together on a line that they cannot be shown. It is also because of this same mechanical difficulty that only the first element of each of the triads of the eighth group is shown. By means of the data given in Table I a skeleton graph may be constructed on which the elements may be placed for which data are not available. The points for such elements are indicated in Figure I by black dots and are determined by plotting the correct values for the term, ln(Z+e), on the appropriate lines. Hence it is possible to compute the density or the atomic volumes of elements not hitherto known if we assume a type of lattice structure. Thus if we ascribe to radium a face-centered cubic lattice similar to the other alkaline earths, the density is found to be about 4.8 instead of the value, “nearly 6” recorded in the literature. Similarly the atomic volume of scandium is not known and according to Figure I it would be 16.7 when a face-centered cubic lattice is assumed. The value approximated by Sommerfeldl from Lothar Meyer’s curve is 18.6. Yo explanation can be offered at this time for the position of the inert gases on this chart. It can only be stated that the location of the points on the appropriate lines can scarcely be fortuitous; nor can the straight line drawn through the points corresponding to these elements be the result of chance. The divergence of the horizontal “group” lines from a straight line becomes less and less as the topmost line is approached, and this fact would indicate that the inert gases form the limit of this property. Figure I does not contain points for any of the ions that do not have the electronic structures of inert gases because the introduction of additional lines would only obscure the striking symmetry exhibited by the figure. Moreover a separate graph is not given because of insufficient data and more especially because these ions do not display such obvious regularities. Nevertheless it is profitable to note a few of the more important characteristics of these ions. When the data for the elements of the first sub-group are plotted, it is possible to draw a straight line through the points of lithium, silver, and gold, the point for copper coming irregularly in the vicinity of those for vanadium and manganese. Likewise for the third sub-group it is interesting to observe that the points for aluminum, gallium, indium, and thallium all lie on a straight line. Finally, if the points for the elements of the fourth group are plotted, it is found that a line passes through the points for carbon, germanium, and tin, whereas lead falls on the line through the points for zirconium and thorium, as is shown in Figure I . Owing to the lack of data and also owing to the fact that these elements which constitute the subSommerfeld: “Atom Structure and Spectral Lines.” Footnote, page 105
310
ARTHUR F. SCOTT
groups frequently exist in several different modifications, it is unwise to make any generalizations. With this brief description of the relationship between the ionic radii and atomic numbers as a starting-point we will now consider those other properties of the elements which are commonly associated with the lattice distances; namely, the compressibility, and the melting point. Quite apart from theoretical grounds it is only necessary to recall the familiar Lothar Meyer curves for these properties in order to perceive their close connection with the radii of ions; and the marked parallel variation of all these properties with the atomic number is sufficient to suggest that, if the radii are related to the , atomic numbers as has been shown, then i-dot13 all of these properties must likewise be ultimately dependent on the atomic number. We may assume that if the proper function could be ascertained, we could obtain a periodic representation for these physical properties similar to Figure I . Unfortunately there are not sufficient data available to test this point although a preliminary examination with the data on hand shows that, by employing the cube root of these properties, representations analogous to Figure I would be obtained. At this time, however, only the data for the alkali metals will be considered because these data are probably the most trustworthy. I n ,~,,,,i~~~~~,~~~~/~ Figure 2 are shown the results obtained Fig. 2 by plotting the values for 2 do, and the The Variation of the Radii, tile cube root of the compressibilities, and the pressibility. and Reciprocal I Melting reciprocal melting points of these elePomt of Alkali Metals. ments against the term, ln(Z+e). The necessary data are given in Table I1 and are taken from Landolt-Bornstein's Tabellen.
''
TABLE I1 The Radii, Compressibilities and Melting Points of the Alkali Metals Element
Cube root of Zompressibility in megabars
(Cuberoot of Reciprocal Melting point
2.08 2.50 3.16 3.50 3.94
13 . o 13.9 14.4 14.7 14.9
x I06
Lithium Sodium Potassium Rubidium Cesium
I ,386 2.485 2.996 3.638 4.025
3 .03 4.00 4.48 4.93 5.34
x IO
ATOMIC NUMBERS AND PROPERTIES O F IONS
311
Apart froin the discrepancy of the point for the radius of cesium which has already been mentioned, the point for potassium is irregular in both of the other cases. This is not a t all disturbing if it is recalled that potassium has no crystalline structure a t high temperatures, a fact which would necessarily cause it to behave abnormally. It is to be noted that with lines 2 and 3 the two branches intersect a t the vertical line corresponding to the value of ln(Z+e) for potassium thus forming triple points which may be taken to represent the probable values for crystalline potassium in both cases. We have therefore these three properties of the alkali metals similarly related to the same function of the atomic numbers and this fact may be considered
I
1 NATURAL )LOGARITHM OF (ZC-ZOI I.5 2.0 2.5 5.0
1
1.5
~
4.0
FIG.3 The Variation of the Lattice Distances of a Series of Halide Salts of the llknli Metals with with the Function In (Zo-Za).
convincing evidence of the validity of the extension of Equation z t o apply to other properties than the radii. Besides the meta,ls there is the large class of polar compounds, the salts, which are characteristically crystalline. It can be seen a t the outset that we have here far more complicated possibilities than exist with the metals, for the atomic number of the negative ion as well as that of the positive ion may vary over a wide range in any specific series of salts. However, to test the applicability of Equation I to such cases we will use the extraordinarily accurate data available for the alkali halides, and plot the distances between the ions against the natural logarithm of absolute value of Z,- Z,, as is shown in Fig. 3. The essential data are given in Table 111.
ARTHUR F. SCOTT
3'2
TABLEI11 The Lattice Distances of the Alkali Halides Fluoride
AZ In AZ Li Na N Rb
6 2 IO
28 ~-Cs4 p-cs 46
1.792 0.693 2.303 3.332 6 3.829 3.829
do
Chloride AZ In AZ
2.009 14 2.311 6 2.680 2 2.85 2 0 2.88 36 2.88 36
2.639 1.792 0.693 2.996 3.584 3.584
do
Bromide AZ In AZ
2.567 32 2.816 24 3.140 16 3.291 2 3.32 20 3.567 2 0
3.446 3.178 2.773 0.693 2.996 2.996
do
2.745 2.982 3.294 3.441 3.51 3.719
AZ
50 42 34 16 2 2
Iodide In A2
tl,
3.912 3.738 3.527 2.773 0.693 0.693
3.007 3.233 3.527 3.668 3.74 3.952
With the exception of the fluorides the values for do are taken from a paper by Fajans and Grimrnl and were in turn calculated by them from the very careful density measurements by Baxter and Wallace2. The values for the fluorides of lithium, sodium and potassium are from determinations by Spangenberg. No measurements of the densities of rubidium and cesium fluorides have been found in the literature; nevertheless, Fajans and Grimm have shown how their molecular volumes may be computed and the data given in the table are obtained in this way, using for the computations the new density determinations of the other fluorides by Spangenbergs. The cesium salts. excepting the fluoride which has the same structure as sodium chloride, have been shown to have a body-centered lattice structure. These salts are given in Table I11 as the &salts and the values for do have been calculated by Grimm from the density measurements of Baxter and Wallace. However, it is net unlikely that these cesium salts, if not some of the other alkali halides, exist jn more than one modification. If so, they would resemble the chemically related ammonium halides which the work of Bartlett and Langmuir4 demonstrated could exist in two modifications; the stable compounds at the lower temperature have the same structure as cesium chloride, and above a definite transition point the structure changes to the rock-salt arrangement. These two types are termed the (3 and ci salts respectively. Further evidence, also of a suggestive nature, is found in the recently published communication of Richards and Saerensj that the fused salts, rudibium bromide and iodide, and cesium iodide, undergo a permanent contraction amounting to somewhat less than one per cent. when they are allowed to stand for a relatively short period of time. This abnormal behavior is readily understandable if it is caused by the transition of the salt crystals from one form to another. If these cesium salts can exist in two modifications, a condition which we will assume, it will be necessary to use the values of do for the a-salts because Fajans and Grimm: 2. Physik., 2, 299 (1920). Baxter and Wallace: J. .4m. Chem. SOC.,38, 259 (1916). 3 Spangenberg: Z. Kryst., 57, 494 (1923). 4Bartlett and Langmuir: J. Am. Chem. SOC.,43, 2, 86 (1921). Richards and Saerens: J. Am. Chem. cfoc., 46, 934 (1924'.
ATOMIC BCMBERS ,4SD PROPERTIES O F IONS
31;
the ions in salts of this type approach nearer than in the /3 salts. Although such salts are purely hypothetical, it is possible to compute values for do by utilizing the relationship of Fajans and Grimml. This has been done, with the molecular volume of cesium fluoride, obtained as previously described, serving as the ba,sis of the calculation and the results are given in Table 111. I n Figure 3 only these values for the hypothetical a-cesium salts are employed in the representation of Equation I ; the circles with the short side-tails indicate the @-saltsand are inserted merely for reference. d word needs to be said concerning the accuracy of these data. Of course, the calculated values are least certain; next to them in the order of uncertainty come the figures for the fluorides, the densities of which are difI
I
I
1
NATURAL LOGAR/ T h M O f (Zc -2%)
1.5
2.0
'2.5
5.0
7.5
Fig. 4 The Variation of the Lattice Distances of a SerieP ?f Alkalihalidic Salts with the Function ln(Z,-Zai.
ficult to determine. The accuracy of the other data would be indisputable were it not for the uncertainty introduced by the abnormal performance noted by Richards and Saerens in the case of salts of rubidium bromide and iodide, and cesium iodide. We therefore cannot have much confidence in the data for these salts. The characteristic features of the various lines in Figure 3 will become apparent through the consideration of two typical lines. We may take first the lines on which the points for the chlorides fall. The change and the reversal in the sign of the constant a a t the point for potassium occurs similarly with all the other lines a t the point where the difference (Z,-Z,) changes sign; that is, wherever the atomic number of the invariant ion in a series is exceeded by the atomic number of the other ion. The other characteristic is the customary discontinuity in the lines a t the potassium ion; in the case of the lines through the chlorides, this second characteristic is masked by the first. Since the distances between ions show a regular variation with the atomic number of the varying cation, it is interesting to compute do by extrapolation 'Fajans and Grimm: Z.Physik. 2, 299
(1920).
314
ARTHUR F. SCOTT
when the anion is in combination with a hypothetical cation of zero atomic number. Since do in all other cases represents the distance at which two ions are in equilibrium to each other, we must interpret it to represent in this special case the distance at which an anion is in equilibrium with a hypothetical cation of zero atomic number; or, it is equal to one half the distance between two similar anions in equilibrium in a crystal. The nearest approach to such a hypothetical situation is to be found with the almost identical analogs, the solid inert elements, where the only difference lies in the nuclear charge which is one unit greater for the inert element than for the corresponding anion. We have already computed (Table I) the distance between the atoms in the solid inert elements on the assumption that they form simple cubic crystals and it will now be interesting to compare these results with the values for 2 do obtained in the present case. The comparison is made below: Chlorine 5.02 Argon 5.26
Bromine Krypton
Iodine Xenon
5.32
5.60
5.82
5.84
Further development of this interesting hypothesis will not be given at this time. Up to this point we have not had occasion to consider any series of compounds in which the negative ion varied while the positive ion remained the same, a case that obtains with the halide salts of each of the alkali metals. An examination of Figure 3 shows the variation of dowith the term, In (2, - 2,) to be entirely different in such cases. However, it is found that by making the constant b depend on the anion of the salt it is possible to bring the points for all the salts of one element on the same line. In practice this is accomplished by assuming a line through the points for the chloride and bromide and by then determining the arbitrary correction for b necessary to bring the points for the fluoride and iodide on the line. These arbitrary corrections are the same in every case and are as follows: Ion
Correction
Fluoride Chloride Bromide Iodide
-0.365
.....
..... +0.170
The results obtained when these corrections are applied are exhibited in Fig: 4. Here as in Fig. 3 the only marked defect in the figure occurs with cesium iodide. I n order to gain assurance that the relationships illustrated in Figures 3 and 4 are not illusory, the linear equations have been established by taking two points on each line as determining points; the values for do for the other points on the line have then been computed from the known value of In (2, Za.) The data for the constants of these linear equations are given inTable IV.
315
I T O M I C NUMBERS AND PROPERTIES O F IOXS
TABLEI V Values for the Constants of Equation Equation 170. I
2
3 4 5 6
7 8 9
Deter mining Points
a - 0.2940 +o ,0626
LiCl - NaCl KC1 - RbCl LiBr - KBr NaI - KI LiCl - LiBr NaC1- NaBr KC1 - KBr RbCl- RbBr CsCl- CsBr
-0.8157 3981 + o . 2 1 52 +o.1198 $0.0740 -0.0651 -0.3197 -I.
I
b
3.343 3.097 5’556 8.458 1.999 2.581 3.089 3.486 4.466
I n Table V we have tabulated the different values for do computed with the aid of these equations. No figures are included which were employed as a basis for determing the constants of the equations. Naturally the corrections for b for the fluoride and iodide salts are applied in the proper equations.
TABLE V The Calculated Distances between Ions compared with Values from Other Sources Salt
LiF LiI NaF NaBr NaI
KC1 KI RbF CsF a-csc1
I
Equation Used
5 5 4 6 3 6 I
7 8 9 2
Calc. Eq. 4 2.020
3.011 2.989 2.299 2.964 3.238 3 . I39 3. 5 2 0 2.90 2.88 3.231
Distances between Ions (do) Bragg’s Densities Empirical X-ray Analysis Table I11 Law Davey Wyckoff
2.019 3.007 3.007 2.322 2.982 3.230 3.140 3.527 2.85 2.88 3.32
2.17 2.90
2.007
2.07
3.537
....
.....
3.03
2.44 2.92 3.13 3.12 3.47 2.92 3.04 3.32
2.310 2.968 3.231 3.138 3.525
2.31 2.98 3.24 3.13 3.55
3.004
3 .OI
.....
....
.....
....
....
I n selecting for the determinant points of the equations upon which the above calculations depend, the values for those salts that lie nearest together on a line, we have the least favorable condition, for an error in either one of the determining points is greatly magnified in the calculated results. Bearing this in mind, the concordance between the values calculated from the equations and those computed from the density measurements (which data should, of course, be the basis of comparison) is entirely satisfactory, ana can be taken as evidence of the validity of the general equation. As a means of further reference, data for do from several other sources are also presented in
316
ARTHUR F. SCOTT
Table V. The data in the fourth column are computed on the assumption that Bragg's empirical law for the additivity of ionic radii is correct, and the values for the radii are taken from his compilation. The last two columns contain the data from measurements by X-ray analysis; the first series contains the values published by Davey' and the second, those by Tyckoff2. There is no reason to believe that the salts of the alkali halides are unique in respect to the relationship between the lattice distance and the atomic numbers of the ions. We would expect the same empirical rule to hold for all other classes of salts. Unfortunately, corresponding accurate data have not been determined for salts other than the alkali halides and therefore the point in question cannot be treated at the present time. Nevertheless it is possible to obtain data for the oxides, sulfides, selenides, and tellurides of the alkaline earths, and representations of these data, similar to Figures 3 and 4, exhibit without exception the identical characteristics which were noted with the alkali halides, when due allowance is made for the greater inaccuracy of the data. Similarly with the same qualification for the uncertainty of the data, the hydrides of the alkali metals behave in a normal fashion. Consequently we must conclude that Equation I is not singularly valid for the pure elements but that it is probably valid for the salts; its limitations cannot 1 I Naf NaCi NaRr Nfll be ascertained at this time. FIG.5 As an outgrowth of the study of the Compressihilities of Alkali Halides metals it was indicated how the various Plotted Against Compressi bilities physical properties might be expressed of the Sodium Halides. by a relationship to the atomic numbers similar to the one found for the radii. This aspect of the problem was suggested by the parallelism of the Lothar Meyer curves for these different properties. The following discussion will be an attempt to demonstrate that a similar parallelism exists for many of the physical properties of the salts, a parallelism which, like the Lothar Meyer curves, must be fundamentally connected with the atomic number. Here, as previously, we will deal with the best known series of salts, the alkali halides.
A number of diverse properties of these salts may be expressed by the general equation : 'Davey: Phys. Rev., 21, 143 (1923). Wyckoff: J. Franklin Inst., 195, 353 (1923).
ATOMIC KCMBERS A S D P R O P E R T I E S O F I O S S
Pat
= CYP,?,
Pay =a
a * y
+P
317
(34
+P
(3b)
where P represents any definite property, c1 and c? represent definite cations in combination with any desired anion, x; and a1 and a2 represent definite anions in salts with a variable cation, y. In both equations a and are constants. The validity of such equations was first pointed out by Fajans and Grimml in the case of the molecular volumes of the alkali halides. Equations in which P was substituted by V, the molecular volume, were found to reproduce the data quantitatively. Another example which will serve also as an illustration of the nature of this relationship is the compressibility. Figure 5 depicts the compressjbilities of the alkali halides plotted against those of the sodium halides.
TABLE VI Compressibilities of the Alkali Halides in Megabars Fluoride
Lithium Sodium Potassium Rubidium Cesium
1.53
Chloride
Bromide
x
106
Iodide
3.7 4.3
5.0 5.3
7.2
(2.6) 3.31
5.2
8.8
(6.1) (3.9)
7.3 5.9
6.4 8.2 7.0
7 . 1
9.3 9.3
The accompanying data given in Table VI with the exception of the fluorides, are those recently published by Richards and Saerens.? The values for lithium and potassium fluoride are taken from determinations by Slater3. The figures for the other fluorides are merely approximations obtained by extrapolation and are enclosed in brackets. These data may be expressed by equations similar to (3a) and (3b) by substituting compressibility for P. The agreement in this case, as can be seen from Figure I , is all that the accuracy of the experimental data warrants. Besides the compressibility there are other properties of these salts which can be represented by Equation 3, and since this fact is not mentioned in the literature, they will be listed here: the contraction in formation from the elements; the contraction on solution; and the molecular refraction of solutions. In addition, Spangenberg has shown three other properties to follow the same rule, namely, the lattice constants of the crystals; the molecular refraction of the crystals; and finally the dispersion, the molecular refraction for different wave-lengths. In the case of this last property, it must be emphasized that only Equation 3a is valid and not 3b. We have thus before us eight separate instances of properties all exhibiting the same regularity and, although this list is probably still incomplete, we have a basis for making a rather obvious general deduction. In no case can 1
Fajans and Grimm: Z. Physik., 2, 299 (1920). Richards and Saerens J. Am. Chem. SOC.,46, 934 (1924). Slater: Phys. Rev., 23, 488 (1924).
318
ARTHCR F. SCOTT
we speak of additive properties, of specific properties of ions. We always find that the property attributable to an individual ion is influenced by the ion with which it is in combination. It has been shown explicitly that the lattice distance of crystals can be expressed as a function of the atomic numbers, and, since the lattice constants are one of the eight cases mentioned above, it must be implied that the other properties can similarly be expressed by some function of the a;tomic numbers of the two ions of a salt.
Summary The present paper must be considered a preliminary attempt to establish a quantitative relationship between the atomic numbers and the properties of ions. It j s shown how the radii of ions in pure elements and the lattice distances of crystals may be expressed with certain qualifications by a linear equation involving the natural logarithm of the absolute difference of the atomic numbers of the ions in a crystal. Furthermore it is indicated how this same empirical relationship may be applied to other physical properties of the metals and salts. The author wishes to take this occasion to express his indebtedness to his friend, Dr. Louis F. Fieser, for many valuable and helpful suggestions during the preparation of this paper.
