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Radii and their changes in the Main-Group elements
3 Radii and their changes in the Main-Group elements The radius of a subshell is the radius, ( r nt) or r max,nl , of the orbital which belongs to this subshell. In the case of strong splitting of p, d and/ orbitals, which occurs in heavy atoms, we use the radii of p 112 , p 312 , d312 , d512 ,f512 andf712 orbitals weighted for closed subshells over their electron occupancy. As the radius of the atom, R, we adopt in this book the radius (rnl) or rmax,nl of the outermost subshell occupied by electrons. The radii of outer subshells determine the experimental covalent, metallic, ionic and van der Waals radii. The radius of the atom and the radii of outer subshells are very important characteristics, because their changes down the Groups affect changes of many chemical properties of respective elements. Although there is a close correlation between the radius of the orbital and its energy (except for the f and 6d orbitals), in this book we mainly use orbital radii to explain properties of elements, because of their more easily visualisable character. 3.1 SCREENING FROM THE NUCLEAR CHARGE Since electrons screen the nuclear attraction, the positive charge acting on a specific electron is not equal to +Ze but +Zene, where Zeff denotes the effecti,·e atomic number gh·en by (3. l) Zeff = Z - S where S is the shielding constant calculated from the Slater rules. The Slater rules are as follows: I. The electronic structure of the atom is written in groupings: (ls) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d, 4.1} and so on. 2. We then assume that: - Electrons in higher orbitals i.e. to the right in the above list, do not shield the nuclear charge for inner electrons, so that their contribution to S is nothing. -
For ns and np valence electrons the contribution to Sis: 0.35 for each electron in the same grouping, except the ls electron which contributes 0.3; 0.85 for each electron in the n-1 grouping; or I for each electron in the n - 2 and lower groupings.
-
For nd and nf valence electrons the contribution from electrons in the same grouping is 0.35 and that from groupings to the left equals 1.0.
[Ch.3
Radii and their changes in the Main-Group elements
22
Slater's rules giYe only approximate \'alues of shielding constants, mainly because penetration of core orbitals by valence electrons is only partly taken into account. Exact \'alues can be obtained from quantum-mechanical calculations. It is interesting to notice the difference between shielding of the nuclear charge for s, p and for d, f electrons by inner electrons. As shown in the above listing, electrons in the n - I grouping contribute 0.85 to shielding constants for s and p electrons and l.O ford and/ electrons. The reason is that d and/ electrons, in contrast to sand p electrons, only negligibly penetrate the ,·icinity of the nucleus (see eq. 1.11) and are, therefore, well screened from the nuclear charge by electrons in the next inner shell. Screening of s and p electrons by the next inner shell is less efficient because they ha,·e a higher probability to reside near the nucleus. As shown by the data in Table 3.1 the effective nuclear charge, from Slater's rules, acting on the Yalence electrons increases as the shells are filled by electrons. The increase of the effective charge across the Period results in a decrease of shell radii, and increase of ionization potentials and electronegativity. The Slater effective nuclear charge acting on the outermost electron increases down each Group, as far as the third row for s-elements and the fourth row for p-elements. Zdr is constant down d-block Groups.
Table 3.1 - Effective nuclear charge, Zen; acting on the outermost s and p electrons. from Slater's rules.
Li
Be
B
c
N
0
F
Ne
l.30
1.95
2.60
3.25
3.90
4.55
5.20
5.85
Na
Mg
Al
Si
p
s
Cl
Ar
2.20
2.85
3.50
4.15
4.80
5.45
6.10
6.75
3.2 CHANGES OF ATOMIC RADII IN s AND p BLOCK ELEMENTS The appearance of a new subshell and its filling with electrons affects the radius of the atom, R, in the following way: l. The beginning of a new shell, i.e. the ns subshell results in a big increase of the atom radius. 2. The radius of the atom also increases when the first electron enters a p subshell, except for the 2p subshell. 3. The appearance of the first d and f electron does not increase the radius of the atom, because d and/ orbitals are inner orbitals. 4. The radius of the atom decreases with filling both outer and inner orbitals by electrons, which is the result of incomplete shielding.
Changes of atomic radii in s and p block elements
Sec. 3.2]
23
360 320
K
280
t:; E
.._a
v
Cs Fr Rb
240
Na
200 160
Xe
120
~e
40 0
Kr
H
80
0
13
2
0
14
15
F
16
17
18
Group Fig. 3.1 The radius of the atom, R = (rout), of s and p block elements. In accordance with its electron configuration helium has been plact:d also in Group 2.
Fig. 3. l shows the radii of the atoms, equal to the radii of the outermost orbitals R
=
(rout> , for sand p block elements. Changes in the radii across Periods and Groups
of the Main Group elements are as follows: l. The radius R decreases between hydrogen and helium due to increasing Zell; and increases significantly with occupying the 2s shell by the first electron at lithium. Due to incomplete shielding the radius R decreases between lithium and beryllium and again between beryllium and boron, in spite of adding the first electron to the 2p shell at boron. The reason is that the effect of increasing Zeff prevails over a small increase due to the appearance of a new subshelL The increase of Zeff across the 2p series makes the radii of atoms from boron to neon decrease significantly. The ,-ery small radii of B, C, N, 0 and F atoms are responsible for unique chemical properties of these elements. For instance, only carbon, nitrogen and ox1·gen form pTC-pTC bonds in homo- and hetero-nuclear molecules such as C=C
N=N
O=O
C=O
N=O
and in the following functional groups: >C=C<
-C=C-
>C=O
-C=N
-N=N-.
Moreover pre-pre bonds also appear in aromatic rings, in graphite, and in fullerenes. Formation of pre-pre bonds requires a small distance between atom centres - such bonding is feasible only between atoms of small radius, see Fig. 3. 2. If atomic
24
Radii and their changes in the Main-Group clements
[Ch. 3
radii increased significantly between beryllium and boron and consequently were
relatively large for succeeding elements, then the diatomic molecules and the functional groups mentioned abO\·e would either be non-existent or would be much less stable. The small atoms carbon, nitrogen and o:-..1·gen can even form n bonds with such much larger atoms of the 3p series as silicon, phosphoms and sulphur. However, the bonds are then not pn-pn but pn-dn. In a pn-dn bond the smaller atom provides the p orbital and two bonding electrons, whereas the larger atom provides the empty d orbital. Because of the shape of the d orbitals, formation of n bonds is feasible, even if the distance between the centres of the atoms is relatively large. see Fig. 3.2. The ability to form pn-dn bonds increases from Si to S, because of the decreasing radius of the atom.
00 pn-pn
overlap
s
s
poor pn-prc overlap
0
s pn-drc
overlap
Fig. 3.2 Correlation between atom size and the fonnation of ;r bonds.
2. In the third Period the radius of the atom decreases between Na and Mg and then increases considerably between Mg and AL in contrast to the decrease between Be and B. The reason is the building up of the 3p subshell, which has a much greater radius than the 3s shell. The increase between Mg and Al results in much greater radii of 3p element atoms than of their lighter congeners. Therefore, the 3p elements only vel}' rarely form n bonds between themselves and with each another. Examples of pn-pn bond formation by 3p elements are bonding in the S = S molecule and in the =P = S functional group. Howe,·er. it should be noticed that recently one of the silicon alkenes R2Si = SiR2 containing a pn-pn bond between the silicon atoms, which are considerably larger than sulphur atoms, has been isolated. Higher radii and lower ionization potentials make the 3p (and also 4p and Sp) elements very different from their lighter congeners. The question arises why the 2p shell, which in contrast lo higher p shells, is a nodeless shell, has an abnormally small size. The explanation which has been ad,·anced is that the 2p, like the 3d and 4/ subshells, has no inner analogues to which it would have to be orthogonal - in other words it does not experience the so-called "primogenic repulsion". A figurative explanation is that p orbitals differ from s orbitals in angular charge distribution, hence repulsion between electrons in 2p and electrons in deeper lying ls and ls orbitals is weak and, consequently, the radius (r 2P) is relatively small. On the other hand the 3p orbitals (and also 4p, 5p and 6p) which have the same symmetry as the inner 2p (3p, 4p and
Sec. 3.2]
Changes of atomic radii in s and p block elements
25
5p) orbitals \vould experience strong repulsion if they had not expanded. This effect is illustrated in Fig. 3.3, which shows a small difference in radial extent between 2s and 2p 112 orbitals and much greater differences between ns and np 112 orbitals for n ~ 3 in Group l3 elements. In the same way (see Fig. 3.1) repulsion between the ls and ls electrons makes the lithium atom much larger than the hydrogen atom (the observed difference is the result of both building a new shell and interelectronic repulsion). One should also notice that when helium, in accordance with its electron configuration, is placed below beryllium in Group 2 then the difference in the radius R between the first and second element in Group 2 becomes large, as is the case with elements of Group 1 and Groups 13 to 18 asssuming that the first element in Group 18 is neon. We shall see in Section 15.6.2 that the radii of d orbitals follow the same pattern.
200
5P112 _ __ 3p1/2 - - -
180
6P112 _ __
4p1/2 _ __
E a. 160
fl
..! 140
3s---
v
120
Ss--4s--
6s---
2P112 - - 2s - - -
100 80 60 40
2p--
20 0
1
3p--
4p--
B
Al
Ga
In
2
3
4
5
Sp--
Tl
6
Period
Fig. 3.3 The ditlt:rence in radial extent between the npl/2 orbitals and ns subshdls in Group 13 elements. 3. Fig. 3.1 shows a big difference in radii between potassium and sodium atoms. The reason is that the 3s shell in Na is built over the small 2p shell of Ne, whereas the 4s shell in K is built over the considerably larger 3p shell of Ar. This is an example of how orbital properties of p block elements are transferred to s block elements. 4. Were it not for the build-up of the 3d subshell, the difference between the radii of gallium and aluminium atoms would be similar to that between potassium and sodium or calcium and magnesium. However, incomplete screening from the nuclear charge by 3d electrons makes the Ga radius even slightly smaller than that
26
Radii and their changes in the Main-Group elements
[Ch. 3
of Al. Generally, the presence of a filled 3d shell makes the -tp elements more similar to their 3p congeners than to other elements in the Group. 5. In the absence of a filled .tf subshell the difference between the radii of thallium and indium atoms should be approximately equal to that between caesium and mbidium. This is because the effect on the radius R of filled .+d and Sd subshells and of building up the Sp and Gp subshells is very similar. However, incomplete screening from the nuclear charge by /electrons decreases the radius of thallium and the elements that follow. An additional factor is the relativistic effect (see Section .+.6) which decreases the radius of the outermost 6p 112 orbital in Tl and Pb. Because of relativistic stabilization of the 6p 112 orbital, which is occupied by one electron in Tl and two electrons in Pb. the radii of the Tl and Pb atoms are even smaller than the radii of the In and Sn atoms. respectively. Both factors, i.e. the presence of the filled f shell and the relativistic effect. result in remarkable similarities between Tl and In and between Pb and Sn. In contrast to the radii of Tl and Pb. those of Bi to Rn are greater than the radii of their homologues in the 5p series. The reason is that.. beginning with Bi.. electrons occupy the 6p 312 orbitals which have a much greater radius than that of the relativistically stabilized 6p 112 orbital. Because splitting of 5p orbitals is much smaller than of Gp orbitals, the radii of Sp elements decrease smoothly from Sn to Xe.
Changes in the radii of d and I subshells are discussed in Chapters lS and 18. respecti' ·ely.
3.3 TYPES OF EXPERIMENTAL RADII 3.3.1 Co\'alent radius The covalent radius for a single bond. r"""' (frequently called the atomic radius) is defined as half the internuclear separation of neighbouring atoms in a singly bonded A 2 molecule as e.g. the Cl 2 molecule. The smaller the internuclear separation (covalent radius) the stronger the cr bond in a homonuclear molecule. Covalent radii of different atoms are usually additive i.e. the following relation holds: d(A-B) = rco\'(A) + rcov(B)
(3.2)
where d(A-B) is the experimentally determined distance between the centres of the A and B atoms; rcov
rcov is a relatively good linear fanction of Y111 ax,out;
-
rco\' is. as expected, very similar to r 111ax.out
27
Ty1>es of ex1>erimental radii
Sec. 3.3]
150 0
I
Se As
Bro
Clo
100
s
0
p
Si
50.
0
50
100
150
Fig. 3.4 Dt:pt:ndc1u:c of tht: covulcnt radius, r"'"., on th.: radius of th.: outt:rmost sht:!l in tht: utom, rmax,out·
3.3.2 Metallic radius The metallic radius, rmet· of an element is defined as equal to half the experimentally determined distance between the nuclei of nearest neighbour atoms in a metallic solid. The smaller the metallic radius the stronger is metallic bonding as measured by atomization energy and, approximately, by melting or boiling temperatures. Since metallic bonding is a highly delocalized form of covalent (electron-deficient) bonding one can also expect a correlation between the internuclear distance and rmax.out· Fig. 3.5 shows the dependence of rmet on rmax,out for a number of metallic elements. One can see that:
-
r111ei depends linearly on rmax.out
-
rmet
is, as expected. approximately equal to rmax,out·
Comparison of Fig. 3.5 with Fig. 3.4 shows almost the same slopes for the two plots, equal to about 0.87. This suggests that with increasing radius of the atom the strengths of both coYalent and metallic bonding decrease at the same rate.
3.3.3 Ionic radius From X-ray or neutron diffraction studies we know only the distance between the nuclei of neighbouring cations and anions. The difficulty lies in dividing such distances into a contribution from the cation, r +• and from the anion, r_. To tackle
Radii and their changes in the Main-Group elements
28
[Ch. 3
this problem one should note that when ions (as a rule anions) of the same element touch each other in an ionic lattice then the radius of the anion is half the internuclear distance. and becomes an experimental value: (3.3)
By changing the cation it is possible to check whether anions are in contact or not. The same M · · · X distance for two different cations means that the x- anions are in contact. The data in Table 3.2 suggest that the 0 2- anions do not touch each another in MnO. but do so in MgO. ll should be noted that the high-spin radius of Mn 2 + is l l pm greater than that of Mg 2+. On the other hand the s2- and Se2- anions are in close contact in the respective sulfides and selenides. It should be noted that the ri (2+) radii of Mg and Mn are very close. The 0 2- anions are in contact in silicates and in spinets. The 0 · · · 0 distance measured in these compounds yields an ri (0 2 -) radius equal to 140 pm. Similar measurements on fluorides result in a value of 133 pm for the radius ri (F-). The rndii of cations are then calculated from the following relations: ri (Mz+) =
r; (Mz+)
d(M · · · 0) - ri (0 2-)
= d(M · · · F) -
ri (F-)
(3.4) (3.5)
250 fmet/pm
200 -
Tl I Lu
M o
n
g o oSc oO Oz Hgo aCd A Hlf,u r 150 GeoAuo c:Q g Ti Sn
Zno
0
oAI
Ga Cu
Fig. 3.5 IJepetH1e111.:e or the metalli<.: radius, atom, '"m.. x.out"
,.met' Oil
the radius or the outennost shell in the
29
T,n>es of experimental radii
Sec. 3.3]
In this method we assume that the ri (0 2-) radius calculated from the 0 · · · 0 distance is the same in all ionic salts i.e. is independent of the cation. We also assume that the radius of the cation calculated from the M · · · 0 distance retains its value in other ionic compounds e.g. in chlorides. These additivity rules hold well only for ions of low charge and large radius. However, the radius of the cation strongly depends on the coordination number, CN, Table 3.3. Table 3.2 - The effect of the cation on the M · · · X distance, in pm. MgO
MnO
MgS
MnS
MgSe
Mn Se
210
224
260
259
273
273
Table 3.3 - Variation of ionic radii of cations (in pm) with coordination number.
CN
Sr2 +
4
Zn 2 +
In 3 +
Zr4 +
60
62
59
6
118
74
80
72
8
126
90
92
84
Despite the doubts and difficulties attendant on the establishment of ionic radii, it should be emphasised that ,·alues on the various empirical scales show remarkably close agreement. The only marked exception is provided by u+, for which values range from 60 to 78 pm. This variation may arise at least in part from uncertainties as to whether it is anion-anion (as in Lil) or anion-cation contact which determines dimensions in various salts LiX. There is a very good linear dependence of the cationic radius of the elements of Groups l, 2, 12, 13 and 14 on the radius of the outermost shell of the respective cations, (rnl,out>· Fig. 3.6. ll can be seen in this Figure that: - for each charge on the cation there is a separate linear relation; - for the same value of (rnl,out> the radius of the cation decreases with increasing charge: - the radius of the cation is as a mle greater than the radius of its outer-most shell. However, the difference between ri and (rnl,out> decreases with increasing charge on the cation:
Radii and their changes in the Main-Group clements
30
[Ch. 3
200 E 175
.e...;-
150 125 100 75 50 25 0
0
20
40
60
80
100
120
Fig. 3.6 Dependern.:e of the ionic radius, ri (CN G), of Group I, 2, D and 14 elements on the radius of the outennost shell in the ion, (rnl,out).
-
Li+ and Be2 + show remarkable deviations, perhaps attributable to their respective outer shells differing from those in their heavier congeners (s in contrast to p subshell). Another explanation is the small radius of the outermost shell in the ion which is 30.3 pm for Li+ and 21.9 pm for Be2+. Such small radii result in shifting electrons from the anion to the cation, which confers some covalent character on the bonding. Covalence decreases the experimental d(M · · · 0) distance, hence the calculated radii of these cations, eq. 3.-l. Similar linear dependences are also shown by cations of d- and .f-block elements (see Figs. 15.3, 15.-l and 18.5). The estimation of "absolute" ionic radii from electron density maps, long after the establishment of the internally consistent and closely comparable sets of empirical ionic radii discussed abm·e, complicated the situation somewhat. The electron density maps, obtained for such salts as LiF, NaCl, KC!, MgO, and CaF 2 , indicated the empirical ionic radii to be systematically in error by approximately 15 pm cations being too small, anions too large. by this amount. A comprehensive set of c1:vsta/ radii was therefore established to take into account the electron density results. it being suggested that crystal radii referred to ions in the crystal lattice, ionic radii to "free ions". The relatively small differences between crystal and ionic
Ty1>es of ex1>erimental radii
Sec. 3.3]
31
radii have a negligible effect on correlations and discussions, except for those involving such quantities as radius ratios where the differences are cumulative. In the present book we use ionic radii throughout, with all values taken from Shannon's extensive tabulation (cf. Preface) where. incidentally, both ionic and crystal radii are listed.
3.3..t Van der Waals radius The van der Waals rnlume, f ·w, is the volume of an atom or a molecule which is not accessible to other atoms (molecules) because of repulsion between nonbonding shells. The energy of repulsion between shells is the consequence of the Pauli principle and strongly decreases with decreasing distance between the centres of the atoms: Erep = BI d12
(3.6)
where B is a constant. Because of the strong dependence of the repulsion energy on the distance, atoms can be considered hard spheres. Atoms which are not bonded chemically attract each other by van der Waals forces so that the total energy, E, becomes: E = BI d12
-
AI
(3.7)
tf6
Fig. 3.7 shows how E depends on the internuclear distance. For atoms of the same element half of the distance d0 is conventionally called the van der Waals radius: (3.8)
E
d
Fig. 3. 7 Dependem:e of energy on the distance between two atoms which do not interact chi:mically.
Radii and their changes in the Main-Group elements
[Ch.3
Since for internuclear distances shorter than d 0 the repulsion energy is a very steep function of d. the van der Waals radius corresponds with very low radial probability density. The rw values are best obtained from X-X contact distances in a solid phase formed by molecules containing chemically bonded X atoms. As seen in Fig. 3 .8. r"" increases with increasing radius of the atom and is larger than R. Comparison of the data in Figs. 3.-l, 3.5 . 3.6 and 3.8 shows that overall the \'arious radii decrease in the follo\\·ing order: 1\,. > R > ,.met > r cov > ri · This order is ,·a!id for elements which form metallic phases and cations. In the case of elements which form anions. the ionic radius is much greater than the covalent and is slightly greater than the van der Waals radius. For instance, in the case of bromine the radii ri. r"., r, 0 ", and H. are I 96, 185, 11-l, and 112 pm respectiYely. It is interesting to note that the ionic radius of a halide ion x- is similar to the van der Waals radius of the isoelectronic noble gas atom. For instance. ri (Br) = 196 pm whereas rw(Kr) = 202 pm.
300
200
0
0
•
rfo <>
Mg
100
e
H
o -
• -
He, Ne, Ar, Kr, Xe
F, Cl, Br, I 0, S, Se, Te x - N, P, As, Sb A - C, Si, Ge, Sn t> - Li, Na, K CJ -
100
200
Rlpm
300
Fig. 3.8 Dependence of the vun
For ns and np valence electrons the contribution to Sis: 0.35 for each electron in the same grouping, except the ls electron which contributes 0.3; 0.85 for each electron in the n-1 grouping; or I for each electron in the n - 2 and lower groupings.
-
For nd and nf valence electrons the contribution from electrons in the same grouping is 0.35 and that from groupings to the left equals 1.0.
[Ch.3
Radii and their changes in the Main-Group elements
22
Slater's rules giYe only approximate \'alues of shielding constants, mainly because penetration of core orbitals by valence electrons is only partly taken into account. Exact \'alues can be obtained from quantum-mechanical calculations. It is interesting to notice the difference between shielding of the nuclear charge for s, p and for d, f electrons by inner electrons. As shown in the above listing, electrons in the n - I grouping contribute 0.85 to shielding constants for s and p electrons and l.O ford and/ electrons. The reason is that d and/ electrons, in contrast to sand p electrons, only negligibly penetrate the ,·icinity of the nucleus (see eq. 1.11) and are, therefore, well screened from the nuclear charge by electrons in the next inner shell. Screening of s and p electrons by the next inner shell is less efficient because they ha,·e a higher probability to reside near the nucleus. As shown by the data in Table 3.1 the effective nuclear charge, from Slater's rules, acting on the Yalence electrons increases as the shells are filled by electrons. The increase of the effective charge across the Period results in a decrease of shell radii, and increase of ionization potentials and electronegativity. The Slater effective nuclear charge acting on the outermost electron increases down each Group, as far as the third row for s-elements and the fourth row for p-elements. Zdr is constant down d-block Groups.
Table 3.1 - Effective nuclear charge, Zen; acting on the outermost s and p electrons. from Slater's rules.
Li
Be
B
c
N
0
F
Ne
l.30
1.95
2.60
3.25
3.90
4.55
5.20
5.85
Na
Mg
Al
Si
p
s
Cl
Ar
2.20
2.85
3.50
4.15
4.80
5.45
6.10
6.75
3.2 CHANGES OF ATOMIC RADII IN s AND p BLOCK ELEMENTS The appearance of a new subshell and its filling with electrons affects the radius of the atom, R, in the following way: l. The beginning of a new shell, i.e. the ns subshell results in a big increase of the atom radius. 2. The radius of the atom also increases when the first electron enters a p subshell, except for the 2p subshell. 3. The appearance of the first d and f electron does not increase the radius of the atom, because d and/ orbitals are inner orbitals. 4. The radius of the atom decreases with filling both outer and inner orbitals by electrons, which is the result of incomplete shielding.
Changes of atomic radii in s and p block elements
Sec. 3.2]
23
360 320
K
280
t:; E
.._a
v
Cs Fr Rb
240
Na
200 160
Xe
120
~e
40 0
Kr
H
80
0
13
2
0
14
15
F
16
17
18
Group Fig. 3.1 The radius of the atom, R = (rout), of s and p block elements. In accordance with its electron configuration helium has been plact:d also in Group 2.
Fig. 3. l shows the radii of the atoms, equal to the radii of the outermost orbitals R
=
(rout> , for sand p block elements. Changes in the radii across Periods and Groups
of the Main Group elements are as follows: l. The radius R decreases between hydrogen and helium due to increasing Zell; and increases significantly with occupying the 2s shell by the first electron at lithium. Due to incomplete shielding the radius R decreases between lithium and beryllium and again between beryllium and boron, in spite of adding the first electron to the 2p shell at boron. The reason is that the effect of increasing Zeff prevails over a small increase due to the appearance of a new subshelL The increase of Zeff across the 2p series makes the radii of atoms from boron to neon decrease significantly. The ,-ery small radii of B, C, N, 0 and F atoms are responsible for unique chemical properties of these elements. For instance, only carbon, nitrogen and ox1·gen form pTC-pTC bonds in homo- and hetero-nuclear molecules such as C=C
N=N
O=O
C=O
N=O
and in the following functional groups: >C=C<
-C=C-
>C=O
-C=N
-N=N-.
Moreover pre-pre bonds also appear in aromatic rings, in graphite, and in fullerenes. Formation of pre-pre bonds requires a small distance between atom centres - such bonding is feasible only between atoms of small radius, see Fig. 3. 2. If atomic
24
Radii and their changes in the Main-Group clements
[Ch. 3
radii increased significantly between beryllium and boron and consequently were
relatively large for succeeding elements, then the diatomic molecules and the functional groups mentioned abO\·e would either be non-existent or would be much less stable. The small atoms carbon, nitrogen and o:-..1·gen can even form n bonds with such much larger atoms of the 3p series as silicon, phosphoms and sulphur. However, the bonds are then not pn-pn but pn-dn. In a pn-dn bond the smaller atom provides the p orbital and two bonding electrons, whereas the larger atom provides the empty d orbital. Because of the shape of the d orbitals, formation of n bonds is feasible, even if the distance between the centres of the atoms is relatively large. see Fig. 3.2. The ability to form pn-dn bonds increases from Si to S, because of the decreasing radius of the atom.
00 pn-pn
overlap
s
s
poor pn-prc overlap
0
s pn-drc
overlap
Fig. 3.2 Correlation between atom size and the fonnation of ;r bonds.
2. In the third Period the radius of the atom decreases between Na and Mg and then increases considerably between Mg and AL in contrast to the decrease between Be and B. The reason is the building up of the 3p subshell, which has a much greater radius than the 3s shell. The increase between Mg and Al results in much greater radii of 3p element atoms than of their lighter congeners. Therefore, the 3p elements only vel}' rarely form n bonds between themselves and with each another. Examples of pn-pn bond formation by 3p elements are bonding in the S = S molecule and in the =P = S functional group. Howe,·er. it should be noticed that recently one of the silicon alkenes R2Si = SiR2 containing a pn-pn bond between the silicon atoms, which are considerably larger than sulphur atoms, has been isolated. Higher radii and lower ionization potentials make the 3p (and also 4p and Sp) elements very different from their lighter congeners. The question arises why the 2p shell, which in contrast lo higher p shells, is a nodeless shell, has an abnormally small size. The explanation which has been ad,·anced is that the 2p, like the 3d and 4/ subshells, has no inner analogues to which it would have to be orthogonal - in other words it does not experience the so-called "primogenic repulsion". A figurative explanation is that p orbitals differ from s orbitals in angular charge distribution, hence repulsion between electrons in 2p and electrons in deeper lying ls and ls orbitals is weak and, consequently, the radius (r 2P) is relatively small. On the other hand the 3p orbitals (and also 4p, 5p and 6p) which have the same symmetry as the inner 2p (3p, 4p and
Sec. 3.2]
Changes of atomic radii in s and p block elements
25
5p) orbitals \vould experience strong repulsion if they had not expanded. This effect is illustrated in Fig. 3.3, which shows a small difference in radial extent between 2s and 2p 112 orbitals and much greater differences between ns and np 112 orbitals for n ~ 3 in Group l3 elements. In the same way (see Fig. 3.1) repulsion between the ls and ls electrons makes the lithium atom much larger than the hydrogen atom (the observed difference is the result of both building a new shell and interelectronic repulsion). One should also notice that when helium, in accordance with its electron configuration, is placed below beryllium in Group 2 then the difference in the radius R between the first and second element in Group 2 becomes large, as is the case with elements of Group 1 and Groups 13 to 18 asssuming that the first element in Group 18 is neon. We shall see in Section 15.6.2 that the radii of d orbitals follow the same pattern.
200
5P112 _ __ 3p1/2 - - -
180
6P112 _ __
4p1/2 _ __
E a. 160
fl
..! 140
3s---
v
120
Ss--4s--
6s---
2P112 - - 2s - - -
100 80 60 40
2p--
20 0
1
3p--
4p--
B
Al
Ga
In
2
3
4
5
Sp--
Tl
6
Period
Fig. 3.3 The ditlt:rence in radial extent between the npl/2 orbitals and ns subshdls in Group 13 elements. 3. Fig. 3.1 shows a big difference in radii between potassium and sodium atoms. The reason is that the 3s shell in Na is built over the small 2p shell of Ne, whereas the 4s shell in K is built over the considerably larger 3p shell of Ar. This is an example of how orbital properties of p block elements are transferred to s block elements. 4. Were it not for the build-up of the 3d subshell, the difference between the radii of gallium and aluminium atoms would be similar to that between potassium and sodium or calcium and magnesium. However, incomplete screening from the nuclear charge by 3d electrons makes the Ga radius even slightly smaller than that
26
Radii and their changes in the Main-Group elements
[Ch. 3
of Al. Generally, the presence of a filled 3d shell makes the -tp elements more similar to their 3p congeners than to other elements in the Group. 5. In the absence of a filled .tf subshell the difference between the radii of thallium and indium atoms should be approximately equal to that between caesium and mbidium. This is because the effect on the radius R of filled .+d and Sd subshells and of building up the Sp and Gp subshells is very similar. However, incomplete screening from the nuclear charge by /electrons decreases the radius of thallium and the elements that follow. An additional factor is the relativistic effect (see Section .+.6) which decreases the radius of the outermost 6p 112 orbital in Tl and Pb. Because of relativistic stabilization of the 6p 112 orbital, which is occupied by one electron in Tl and two electrons in Pb. the radii of the Tl and Pb atoms are even smaller than the radii of the In and Sn atoms. respectively. Both factors, i.e. the presence of the filled f shell and the relativistic effect. result in remarkable similarities between Tl and In and between Pb and Sn. In contrast to the radii of Tl and Pb. those of Bi to Rn are greater than the radii of their homologues in the 5p series. The reason is that.. beginning with Bi.. electrons occupy the 6p 312 orbitals which have a much greater radius than that of the relativistically stabilized 6p 112 orbital. Because splitting of 5p orbitals is much smaller than of Gp orbitals, the radii of Sp elements decrease smoothly from Sn to Xe.
Changes in the radii of d and I subshells are discussed in Chapters lS and 18. respecti' ·ely.
3.3 TYPES OF EXPERIMENTAL RADII 3.3.1 Co\'alent radius The covalent radius for a single bond. r"""' (frequently called the atomic radius) is defined as half the internuclear separation of neighbouring atoms in a singly bonded A 2 molecule as e.g. the Cl 2 molecule. The smaller the internuclear separation (covalent radius) the stronger the cr bond in a homonuclear molecule. Covalent radii of different atoms are usually additive i.e. the following relation holds: d(A-B) = rco\'(A) + rcov(B)
(3.2)
where d(A-B) is the experimentally determined distance between the centres of the A and B atoms; rcov
rcov is a relatively good linear fanction of Y111 ax,out;
-
rco\' is. as expected, very similar to r 111ax.out
27
Ty1>es of ex1>erimental radii
Sec. 3.3]
150 0
I
Se As
Bro
Clo
100
s
0
p
Si
50.
0
50
100
150
Fig. 3.4 Dt:pt:ndc1u:c of tht: covulcnt radius, r"'"., on th.: radius of th.: outt:rmost sht:!l in tht: utom, rmax,out·
3.3.2 Metallic radius The metallic radius, rmet· of an element is defined as equal to half the experimentally determined distance between the nuclei of nearest neighbour atoms in a metallic solid. The smaller the metallic radius the stronger is metallic bonding as measured by atomization energy and, approximately, by melting or boiling temperatures. Since metallic bonding is a highly delocalized form of covalent (electron-deficient) bonding one can also expect a correlation between the internuclear distance and rmax.out· Fig. 3.5 shows the dependence of rmet on rmax,out for a number of metallic elements. One can see that:
-
r111ei depends linearly on rmax.out
-
rmet
is, as expected. approximately equal to rmax,out·
Comparison of Fig. 3.5 with Fig. 3.4 shows almost the same slopes for the two plots, equal to about 0.87. This suggests that with increasing radius of the atom the strengths of both coYalent and metallic bonding decrease at the same rate.
3.3.3 Ionic radius From X-ray or neutron diffraction studies we know only the distance between the nuclei of neighbouring cations and anions. The difficulty lies in dividing such distances into a contribution from the cation, r +• and from the anion, r_. To tackle
Radii and their changes in the Main-Group elements
28
[Ch. 3
this problem one should note that when ions (as a rule anions) of the same element touch each other in an ionic lattice then the radius of the anion is half the internuclear distance. and becomes an experimental value: (3.3)
By changing the cation it is possible to check whether anions are in contact or not. The same M · · · X distance for two different cations means that the x- anions are in contact. The data in Table 3.2 suggest that the 0 2- anions do not touch each another in MnO. but do so in MgO. ll should be noted that the high-spin radius of Mn 2 + is l l pm greater than that of Mg 2+. On the other hand the s2- and Se2- anions are in close contact in the respective sulfides and selenides. It should be noted that the ri (2+) radii of Mg and Mn are very close. The 0 2- anions are in contact in silicates and in spinets. The 0 · · · 0 distance measured in these compounds yields an ri (0 2 -) radius equal to 140 pm. Similar measurements on fluorides result in a value of 133 pm for the radius ri (F-). The rndii of cations are then calculated from the following relations: ri (Mz+) =
r; (Mz+)
d(M · · · 0) - ri (0 2-)
= d(M · · · F) -
ri (F-)
(3.4) (3.5)
250 fmet/pm
200 -
Tl I Lu
M o
n
g o oSc oO Oz Hgo aCd A Hlf,u r 150 GeoAuo c:Q g Ti Sn
Zno
0
oAI
Ga Cu
Fig. 3.5 IJepetH1e111.:e or the metalli<.: radius, atom, '"m.. x.out"
,.met' Oil
the radius or the outennost shell in the
29
T,n>es of experimental radii
Sec. 3.3]
In this method we assume that the ri (0 2-) radius calculated from the 0 · · · 0 distance is the same in all ionic salts i.e. is independent of the cation. We also assume that the radius of the cation calculated from the M · · · 0 distance retains its value in other ionic compounds e.g. in chlorides. These additivity rules hold well only for ions of low charge and large radius. However, the radius of the cation strongly depends on the coordination number, CN, Table 3.3. Table 3.2 - The effect of the cation on the M · · · X distance, in pm. MgO
MnO
MgS
MnS
MgSe
Mn Se
210
224
260
259
273
273
Table 3.3 - Variation of ionic radii of cations (in pm) with coordination number.
CN
Sr2 +
4
Zn 2 +
In 3 +
Zr4 +
60
62
59
6
118
74
80
72
8
126
90
92
84
Despite the doubts and difficulties attendant on the establishment of ionic radii, it should be emphasised that ,·alues on the various empirical scales show remarkably close agreement. The only marked exception is provided by u+, for which values range from 60 to 78 pm. This variation may arise at least in part from uncertainties as to whether it is anion-anion (as in Lil) or anion-cation contact which determines dimensions in various salts LiX. There is a very good linear dependence of the cationic radius of the elements of Groups l, 2, 12, 13 and 14 on the radius of the outermost shell of the respective cations, (rnl,out>· Fig. 3.6. ll can be seen in this Figure that: - for each charge on the cation there is a separate linear relation; - for the same value of (rnl,out> the radius of the cation decreases with increasing charge: - the radius of the cation is as a mle greater than the radius of its outer-most shell. However, the difference between ri and (rnl,out> decreases with increasing charge on the cation:
Radii and their changes in the Main-Group clements
30
[Ch. 3
200 E 175
.e...;-
150 125 100 75 50 25 0
0
20
40
60
80
100
120
Fig. 3.6 Dependern.:e of the ionic radius, ri (CN G), of Group I, 2, D and 14 elements on the radius of the outennost shell in the ion, (rnl,out).
-
Li+ and Be2 + show remarkable deviations, perhaps attributable to their respective outer shells differing from those in their heavier congeners (s in contrast to p subshell). Another explanation is the small radius of the outermost shell in the ion which is 30.3 pm for Li+ and 21.9 pm for Be2+. Such small radii result in shifting electrons from the anion to the cation, which confers some covalent character on the bonding. Covalence decreases the experimental d(M · · · 0) distance, hence the calculated radii of these cations, eq. 3.-l. Similar linear dependences are also shown by cations of d- and .f-block elements (see Figs. 15.3, 15.-l and 18.5). The estimation of "absolute" ionic radii from electron density maps, long after the establishment of the internally consistent and closely comparable sets of empirical ionic radii discussed abm·e, complicated the situation somewhat. The electron density maps, obtained for such salts as LiF, NaCl, KC!, MgO, and CaF 2 , indicated the empirical ionic radii to be systematically in error by approximately 15 pm cations being too small, anions too large. by this amount. A comprehensive set of c1:vsta/ radii was therefore established to take into account the electron density results. it being suggested that crystal radii referred to ions in the crystal lattice, ionic radii to "free ions". The relatively small differences between crystal and ionic
Ty1>es of ex1>erimental radii
Sec. 3.3]
31
radii have a negligible effect on correlations and discussions, except for those involving such quantities as radius ratios where the differences are cumulative. In the present book we use ionic radii throughout, with all values taken from Shannon's extensive tabulation (cf. Preface) where. incidentally, both ionic and crystal radii are listed.
3.3..t Van der Waals radius The van der Waals rnlume, f ·w, is the volume of an atom or a molecule which is not accessible to other atoms (molecules) because of repulsion between nonbonding shells. The energy of repulsion between shells is the consequence of the Pauli principle and strongly decreases with decreasing distance between the centres of the atoms: Erep = BI d12
(3.6)
where B is a constant. Because of the strong dependence of the repulsion energy on the distance, atoms can be considered hard spheres. Atoms which are not bonded chemically attract each other by van der Waals forces so that the total energy, E, becomes: E = BI d12
-
AI
(3.7)
tf6
Fig. 3.7 shows how E depends on the internuclear distance. For atoms of the same element half of the distance d0 is conventionally called the van der Waals radius: (3.8)
E
d
Fig. 3. 7 Dependem:e of energy on the distance between two atoms which do not interact chi:mically.
Radii and their changes in the Main-Group elements
[Ch.3
Since for internuclear distances shorter than d 0 the repulsion energy is a very steep function of d. the van der Waals radius corresponds with very low radial probability density. The rw values are best obtained from X-X contact distances in a solid phase formed by molecules containing chemically bonded X atoms. As seen in Fig. 3 .8. r"" increases with increasing radius of the atom and is larger than R. Comparison of the data in Figs. 3.-l, 3.5 . 3.6 and 3.8 shows that overall the \'arious radii decrease in the follo\\·ing order: 1\,. > R > ,.met > r cov > ri · This order is ,·a!id for elements which form metallic phases and cations. In the case of elements which form anions. the ionic radius is much greater than the covalent and is slightly greater than the van der Waals radius. For instance, in the case of bromine the radii ri. r"., r, 0 ", and H. are I 96, 185, 11-l, and 112 pm respectiYely. It is interesting to note that the ionic radius of a halide ion x- is similar to the van der Waals radius of the isoelectronic noble gas atom. For instance. ri (Br) = 196 pm whereas rw(Kr) = 202 pm.
300
200
0
0
•
rfo <>
Mg
100
e
H
o -
• -
He, Ne, Ar, Kr, Xe
F, Cl, Br, I 0, S, Se, Te x - N, P, As, Sb A - C, Si, Ge, Sn t> - Li, Na, K CJ -
100
200
Rlpm
300
Fig. 3.8 Dependence of the vun