Ionic radius

Ionic radius, rion, is the radius of an atom's ion in ionic crystals structure. Although neither atoms nor ions have sharp boundaries, they are sometimes treated as if they were hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice. Ionic radii are typically given in units of either picometers (pm) or angstroms (Å), with 1 Å = 100 pm. Typical values range from 31 pm (0.3 Å) to over 200 pm (2 Å).

The concept can be extended to solvated ions in liquid solutions taking into consideration the solvation shell.

X NaX AgX
F 464 492
Cl 564 555
Br 598 577
Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. All compounds crystallize in the NaCl structure.

Ions may be larger or smaller than the neutral atom, depending on the ion's electric charge. When an atom loses an electron to form a cation, the other electrons are more attracted to the nucleus, and the radius of the ion gets smaller. Similarly, when an electron is added to an atom, forming an anion, the added electron increases the size of the electron cloud by interelectronic repulsion.

The ionic radius is not a fixed property of a given ion, but varies with coordination number, spin state and other parameters. Nevertheless, ionic radius values are sufficiently transferable to allow periodic trends to be recognized. As with other types of atomic radius, ionic radii increase on descending a group. Ionic size (for the same ion) also increases with increasing coordination number, and an ion in a high-spin state will be larger than the same ion in a low-spin state. In general, ionic radius decreases with increasing positive charge and increases with increasing negative charge.

An "anomalous" ionic radius in a crystal is often a sign of significant covalent character in the bonding. No bond is completely ionic, and some supposedly "ionic" compounds, especially of the transition metals, are particularly covalent in character. This is illustrated by the unit cell parameters for sodium and silver halides in the table. On the basis of the fluorides, one would say that Ag+ is larger than Na+, but on the basis of the chlorides and bromides the opposite appears to be true.[1] This is because the greater covalent character of the bonds in AgCl and AgBr reduces the bond length and hence the apparent ionic radius of Ag+, an effect which is not present in the halides of the more electropositive sodium, nor in silver fluoride in which the fluoride ion is relatively unpolarizable.

Determination

The distance between two ions in an ionic crystal can be determined by X-ray crystallography, which gives the lengths of the sides of the unit cell of a crystal. For example, the length of each edge of the unit cell of sodium chloride is found to be 564.02 pm. Each edge of the unit cell of sodium chloride may be considered to have the atoms arranged as Na+∙∙∙Cl∙∙∙Na+, so the edge is twice the Na-Cl separation. Therefore, the distance between the Na+ and Cl ions is half of 564.02 pm, which is 282.01 pm. However, although X-ray crystallography gives the distance between ions, it doesn't indicate where the boundary is between those ions, so it doesn't directly give ionic radii.

Landé[2] estimated ionic radii by considering crystals in which the anion and cation have a large difference in size, such as LiI. The lithium ions are so much smaller than the iodide ions that the lithium fits into holes within the crystal lattice, allowing the iodide ions to touch. That is, the distance between two neighboring iodides in the crystal is assumed to be twice the radius of the iodide ion, which was deduced to be 214 pm. This value can be used to determine other radii. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. In this way values for the radii of 8 ions were determined.

Wasastjerna estimated ionic radii by considering the relative volumes of ions as determined from electrical polarizability as determined by measurements of refractive index.[3] These results were extended by Victor Goldschmidt.[4] Both Wasastjerna and Goldschmidt used a value of 132 pm for the O2− ion.

Pauling used effective nuclear charge to proportion the distance between ions into anionic and a cationic radii.[5] His data gives the O2− ion a radius of 140 pm.

A major review of crystallographic data led to the publication of revised ionic radii by Shannon.[6] Shannon gives different radii for different coordination numbers, and for high and low spin states of the ions. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as "effective" ionic radii. However, Shannon also includes data based on rion(O2−) = 126 pm; data using that value are referred to as "crystal" ionic radii. Shannon states that "it is felt that crystal radii correspond more closely to the physical size of ions in a solid."[6] The two sets of data are listed in the two tables below.

Crystal ionic radii in pm of elements in function of ionic charge and spin
(ls = low spin, hs= high spin).
Ions are 6-coordinate unless indicated differently in parentheses
(e.g. 146 (4) for 4-coordinate N3−).[6]
Number Name Symbol 3– 2– 1– 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+
1 Hydrogen H −4 (2)
3 Lithium Li 90
4 Beryllium Be 59
5 Boron B 41
6 Carbon C 30
7 Nitrogen N 132 (4)3027
8 Oxygen O 126
9 Fluorine F 11922
11 Sodium Na 116
12 Magnesium Mg 86
13 Aluminium Al 67.5
14 Silicon Si 54
15 Phosphorus P 5852
16 Sulfur S 1705143
17 Chlorine Cl 16726 (3py)41
19 Potassium K 152
20 Calcium Ca 114
21 Scandium Sc 88.5
22 Titanium Ti 1008174.5
23 Vanadium V 93787268
24 Chromium ls Cr 8775.5696358
24 Chromium hs Cr 94
25 Manganese ls Mn 81726747 (4)39.5 (4)60
25 Manganese hs Mn 9778.5
26 Iron ls Fe 756972.539 (4)
26 Iron hs Fe 9278.5
27 Cobalt ls Co 7968.5
27 Cobalt hs Co 88.57567
28 Nickel hs Ni 837062 ls
28 Nickel ls Ni 74
29 Copper Cu 918768 ls
30 Zinc Zn 88
31 Gallium Ga 76
32 Germanium Ge 8767
33 Arsenic As 7260
34 Selenium Se 1846456
35 Bromine Br 18273 (4sq)45 (3py)53
37 Rubidium Rb 166
38 Strontium Sr 132
39 Yttrium Y 104
40 Zirconium Zr 86
41 Niobium Nb 868278
42 Molybdenum Mo 83797573
43 Technetium Tc 78.57470
44 Ruthenium Ru 827670.552 (4)50 (4)
45 Rhodium Rh 80.57469
46 Palladium Pd 73 (2)1009075.5
47 Silver Ag 12910889
48 Cadmium Cd 109
49 Indium In 94
50 Tin Sn 83
51 Antimony Sb 9074
52 Tellurium Te 20711170
53 Iodine I 20610967
54 Xenon Xe 62
55 Caesium Cs 181
56 Barium Ba 149
57 Lanthanum La 117.2
58 Cerium Ce 115101
59 Praseodymium Pr 11399
60 Neodymium Nd 143 (8)112.3
61 Promethium Pm 111
62 Samarium Sm 136 (7)109.8
63 Europium Eu 131108.7
64 Gadolinium Gd 107.8
65 Terbium Tb 106.390
66 Dysprosium Dy 121105.2
67 Holmium Ho 104.1
68 Erbium Er 103
69 Thulium Tm 117102
70 Ytterbium Yb 116100.8
71 Lutetium Lu 100.1
72 Hafnium Hf 85
73 Tantalum Ta 868278
74 Tungsten W 807674
75 Rhenium Re 77726967
76 Osmium Os 7771.568.566.553 (4)
77 Iridium Ir 8276.571
78 Platinum Pt 9476.571
79 Gold Au 1519971
80 Mercury Hg 133116
81 Thallium Tl 164102.5
82 Lead Pb 13391.5
83 Bismuth Bi 11790
84 Polonium Po 10881
85 Astatine At 76
87 Francium Fr 194
88 Radium Ra 162 (8)
89 Actinium Ac 126
90 Thorium Th 108
91 Protactinium Pa 11610492
92 Uranium U 116.51039087
93 Neptunium Np 124115101898685
94 Plutonium Pu 1141008885
95 Americium Am 140 (8)111.599
96 Curium Cm 11199
97 Berkelium Bk 11097
98 Californium Cf 10996.1
99 Einsteinium Es 92.8[7]
Effective ionic radii in pm of elements in function of ionic charge and spin
(ls = low spin, hs= high spin).
Ions are 6-coordinate unless indicated differently in parentheses
(e.g. 146 (4) for 4-coordinate N3−).[6]
Number Name Symbol 3– 2– 1– 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+
1 Hydrogen H −18 (2)
3 Lithium Li 76
4 Beryllium Be 45
5 Boron B 27
6 Carbon C 16
7 Nitrogen N 146 (4)1613
8 Oxygen O 140
9 Fluorine F 1338
11 Sodium Na 102
12 Magnesium Mg 72
13 Aluminium Al 53.5
14 Silicon Si 40
15 Phosphorus P 4438
16 Sulfur S 1843729
17 Chlorine Cl 18112 (3py)27
19 Potassium K 138
20 Calcium Ca 100
21 Scandium Sc 74.5
22 Titanium Ti 866760.5
23 Vanadium V 79645854
24 Chromium ls Cr 7361.5554944
24 Chromium hs Cr 80
25 Manganese ls Mn 67585333 (4)25.5 (4)46
25 Manganese hs Mn 8364.5
26 Iron ls Fe 615558.525 (4)
26 Iron hs Fe 7864.5
27 Cobalt ls Co 6554.5
27 Cobalt hs Co 74.56153 hs
28 Nickel ls Ni 695648 ls
28 Nickel hs Ni 60
29 Copper Cu 777354 ls
30 Zinc Zn 74
31 Gallium Ga 62
32 Germanium Ge 7353
33 Arsenic As 5846
34 Selenium Se 1985042
35 Bromine Br 19659 (4sq)31 (3py)39
37 Rubidium Rb 152
38 Strontium Sr 118
39 Yttrium Y 90
40 Zirconium Zr 72
41 Niobium Nb 726864
42 Molybdenum Mo 69656159
43 Technetium Tc 64.56056
44 Ruthenium Ru 686256.538 (4)36 (4)
45 Rhodium Rh 66.56055
46 Palladium Pd 59 (2)867661.5
47 Silver Ag 1159475
48 Cadmium Cd 95
49 Indium In 80
50 Tin Sn 69
51 Antimony Sb 7660
52 Tellurium Te 2219756
53 Iodine I 2209553
54 Xenon Xe 48
55 Caesium Cs 167
56 Barium Ba 135
57 Lanthanum La 103.2
58 Cerium Ce 10187
59 Praseodymium Pr 9985
60 Neodymium Nd 129 (8)98.3
61 Promethium Pm 97
62 Samarium Sm 122 (7)95.8
63 Europium Eu 11794.7
64 Gadolinium Gd 93.5
65 Terbium Tb 92.376
66 Dysprosium Dy 10791.2
67 Holmium Ho 90.1
68 Erbium Er 89
69 Thulium Tm 10388
70 Ytterbium Yb 10286.8
71 Lutetium Lu 86.1
72 Hafnium Hf 71
73 Tantalum Ta 726864
74 Tungsten W 666260
75 Rhenium Re 63585553
76 Osmium Os 6357.554.552.539 (4)
77 Iridium Ir 6862.557
78 Platinum Pt 8062.557
79 Gold Au 1378557
80 Mercury Hg 119102
81 Thallium Tl 15088.5
82 Lead Pb 11977.5
83 Bismuth Bi 10376
84 Polonium Po 9467
85 Astatine At 62
87 Francium Fr 180
88 Radium Ra 148 (8)
89 Actinium Ac 112
90 Thorium Th 94
91 Protactinium Pa 1049078
92 Uranium U 102.5897673
93 Neptunium Np 11010187757271
94 Plutonium Pu 100867471
95 Americium Am 126 (8)97.585
96 Curium Cm 9785
97 Berkelium Bk 9683
98 Californium Cf 9582.1
99 Einsteinium Es 83.5[7]

Soft-sphere model

Soft-sphere ionic radii (in pm) of some ions
Cation, M RM Anion, X RX
Li+ 109.4 Cl 218.1
Na+ 149.7 Br 237.2

For many compounds, the model of ions as hard spheres does not reproduce the distance between ions, , to the accuracy with which it can be measured in crystals. One approach to improving the calculated accuracy is to model ions as "soft spheres" that overlap in the crystal. Because the ions overlap, their separation in the crystal will be less than the sum of their soft-sphere radii.[8]

The relation between soft-sphere ionic radii, and , and , is given by

,

where is an exponent that varies with the type of crystal structure. In the hard-sphere model, would be 1, giving .

Comparison between observed and calculated ion separations (in pm)
MX Observed Soft-sphere model
LiCl 257.0 257.2
LiBr 275.1 274.4
NaCl 282.0 281.9
NaBr 298.7 298.2

In the soft-sphere model, has a value between 1 and 2. For example, for crystals of group 1 halides with the sodium chloride structure, a value of 1.6667 gives good agreement with experiment. Some soft-sphere ionic radii are in the table. These radii are larger than the crystal radii given above (Li+, 90 pm; Cl, 167 pm). Inter-ionic separations calculated with these radii give remarkably good agreement with experimental values. Some data are given in the table. Curiously, no theoretical justification for the equation containing has been given.

Non-spherical ions

The concept of ionic radii is based on the assumption of a spherical ion shape. However, from a group-theoretical point of view the assumption is only justified for ions that reside on high-symmetry crystal lattice sites like Na and Cl in halite or Zn and S in sphalerite. A clear distinction can be made, when the point symmetry group of the respective lattice site is considered,[9] which are the cubic groups Oh and Td in NaCl and ZnS. For ions on lower-symmetry sites significant deviations of their electron density from a spherical shape may occur. This holds in particular for ions on lattice sites of polar symmetry, which are the crystallographic point groups C1, C1h, Cn or Cnv, n = 2, 3, 4 or 6.[10] A thorough analysis of the bonding geometry was recently carried out for pyrite-type compounds, where monovalent chalcogen ions reside on C3 lattice sites. It was found that chalcogen ions have to be modeled by ellipsoidal charge distributions with different radii along the symmetry axis and perpendicular to it.[11]

See also

References

  1. On the basis of conventional ionic radii, Ag+ (129 pm) is indeed larger than Na+ (116 pm)
  2. Landé, A. (1920). "Über die Größe der Atome". Zeitschrift für Physik. 1 (3): 191–197. Bibcode:1920ZPhy....1..191L. doi:10.1007/BF01329165. Retrieved 1 June 2011.
  3. Wasastjerna, J. A. (1923). "On the radii of ions". Comm. Phys.-Math., Soc. Sci. Fenn. 1 (38): 1–25.
  4. Goldschmidt, V. M. (1926). Geochemische Verteilungsgesetze der Elemente. Skrifter Norske Videnskaps—Akad. Oslo, (I) Mat. Natur. This is an 8 volume set of books by Goldschmidt.
  5. Pauling, L. (1960). The Nature of the Chemical Bond (3rd Edn.). Ithaca, NY: Cornell University Press.
  6. R. D. Shannon (1976). "Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides". Acta Crystallogr A. 32 (5): 751–767. Bibcode:1976AcCrA..32..751S. doi:10.1107/S0567739476001551.
  7. R. G. Haire, R. D. Baybarz: "Identification and Analysis of Einsteinium Sesquioxide by Electron Diffraction", in: Journal of Inorganic and Nuclear Chemistry, 1973, 35 (2), S. 489–496; doi:10.1016/0022-1902(73)80561-5.
  8. Lang, Peter F.; Smith, Barry C. (2010). "Ionic radii for Group 1 and Group 2 halide, hydride, fluoride, oxide, sulfide, selenide and telluride crystals". Dalton Transactions. 39 (33): 7786–7791. doi:10.1039/C0DT00401D. PMID 20664858.
  9. H. Bethe (1929). "Termaufspaltung in Kristallen". Annalen der Physik. 3 (2): 133–208. Bibcode:1929AnP...395..133B. doi:10.1002/andp.19293950202.
  10. M. Birkholz (1995). "Crystal-field induced dipoles in heteropolar crystals – I. concept". Z. Phys. B. 96 (3): 325–332. Bibcode:1995ZPhyB..96..325B. CiteSeerX 10.1.1.424.5632. doi:10.1007/BF01313054.
  11. M. Birkholz (2014). "Modeling the Shape of Ions in Pyrite-Type Crystals". Crystals. 4 (3): 390–403. doi:10.3390/cryst4030390.
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