Abstract
The valencies of the lanthanides vary more than was once thought. In addition to valencies associated with a half-full shell, there are valencies associated with a quarter- and three-quarter-full shell. This can be explained on the basis of Slater’s theory of many-electron atoms. The same theory explains the variation in complexing constants in the trivalent state (the “tetrad effect”). Valency in metallic and organometallic compounds is also discussed.
Keywords
Lanthanide Valency f shell Ionization energy Hydration energy Coulomb energy Exchange energy Nephelauxetic effect Tetrad effectIntroduction
Fifty years ago, the lanthanides1 were little more than a chemical curiosity. They were thought of as being very similar, and placed together in the same box of the Periodic Table. Since then, their chemistry has been studied in more detail, and found to vary more than was once thought. Also, some now have important applications, e.g. neodymium in lasers and magnets.
Discussions of lanthanide compounds and ions are often conducted using a classification by oxidation state. However, certain distinctions that we make in this paper are not well served by this concept. For example, some lanthanide monosulphides, LnS, are metallic and others are semiconductors, but in both types of compound, the formal oxidation state of the lanthanide is +2. Here, therefore, we make the valency of the lanthanide element our classifying instrument and extend the classical definition of valency (Nelson 1997) to metals and metallic compounds by adding the number of electrons contributed to the metallic bonding. This gives sodium, for example, the same valency in the metal (Na+e−) as in its compounds (NaF, NaCl, etc.). An example more relevant to this paper is LaI2 in which the lanthanum atom has an oxidation number of +2. The compound, however, is bronze in colour, has a high electrical conductivity, and is diamagnetic [La2+(I−)2 would be paramagnetic]. It is accordingly formulated La3+(I−)2e−, in which the lanthanum has a valency of three. This definition of valency is essentially that adopted by Grimm and Sommerfeld (1926) and endorsed by Sidgwick (1927: 182): “it is numerically equal to the number of electrons of the atom ‘engaged’ (beansprucht) in attaching the other atoms”.
The variation in the valencies of the elements across the lanthanide series is of particular interest. Here we present a simple discussion of this problem based largely upon work which we have published elsewhere (Johnson 1969a, b, c, 1974, 1977, 1980, 1982a, b, 2006; Johnson and Corbett 1970).
Principal valency
Valencies of the lanthanides
Valencies | ||||
---|---|---|---|---|
La | 3 | |||
Ce | 3 | 4 | ||
Pr | 3 | 4 | (5)b | |
Nd | 2 | 3 | 4 | |
Pm | (2)a | 3 | ||
Sm | 2 | 3 | ||
Eu | 2 | 3 | ||
Gd | 3 | |||
Tb | 3 | 4 | ||
Dy | 2 | 3 | 4 | |
Ho | 3 | |||
Er | 3 | |||
Tm | 2 | 3 | (4?)c | |
Yb | 2 | 3 | ||
Lu | 3 |
Ln–O distances in [Ln(H2O)9](C2H5OSO3)3 salts (Gerkin and Reppart 1984) plotted against number of f electrons
The first term in Eq. (3) is approximately equal to the Coulomb energy between the electrons (J), and the second term to the exchange energy between electrons of the same spin (K).4 This arises because electrons can keep apart better if their spins are parallel rather than antiparallel. The third term is associated with the ability of electrons to keep apart better if they are travelling round a nucleus in the same direction rather than in opposite directions. This lowers the repulsion between electrons whose m l values have the same sign compared with those whose m l values have the opposite sign.5
Table 2 also shows that δq and c 3 bottom out at f3 and f4 and at f10 and f11. Since the terms in q and c 3 in Eq. (4) have opposite signs, the effect of this bottoming out will depend on the relative values of ΔE q and ΔE 3. Although ΔE 3/ΔE q is less than one, if it is sufficiently large, the variation in c 3 will predominate because of the large negative values at f3, f4, f10, and f11. In this case, when there is shell expansion (ΔE 3 < 0), the values of ln K eq will bottom out in the same way. This accords with Fig. 2, which can be reproduced with ΔE q = −36 cm−1 and ΔE 3 = −7 cm−1, coupled with a linear increase. A more precise analysis takes account of changes in spin-orbit coupling and ligand-field stabilization.
A secondary effect, which is not brought out by Eq. (3), is the higher repulsion exercised between electrons in the same orbital as compared with different orbitals (Blake 1981). We discuss this elsewhere (see footnote 4).
Other valencies
Configurations of ions corresponding to the valencies in Table 1
Hydration enthalpies of Ln3+ ions (h 3) plotted against number of f electrons
Ionization enthalpies of Ln3+ ions (i 4) plotted against number of f electrons
Minus the ionization enthalpies of Ln2+ ions (enthalpies of Ln3+ + e− → Ln2+) plotted against number of outer electrons. Blue points, f n−1 → f n ; red points, f n−1 → f n−1d1 for ions having a f n−1d1 ground state (La2+, d1; Gd2+, f7d1)
The marked irregularity around the three-quarter shell is also apparent in Fig. 4. It suggests that compounds of tetravalent thulium, such as Cs3TmF7, might be unexpectedly stable. Some workers claim to have prepared impure samples of this compound (Spitsin et al. 1982), but others have reported little or no oxidation [unpublished observations by the authors (1972), L. B. Asprey (1975) and R. Hoppe (1985)] and further confirmation is needed. This explains the question marks in Tables 1 and 3.
The reason for the big break in i 3 and i 4 is well known from the transition series. Repulsion energies between electrons are reduced by the exchange energy between electrons with the same spin. The latter builds up until a shell is half-full, then drops to zero as an electron is added with opposite spin. It then builds up again as more electrons are added with this spin.
Values of the coefficients in Eq. (6)
Process | Δi p | Δi q | Δi c 3 |
---|---|---|---|
f1 → f0 | 0 | 0 | 0 |
f2 → f1 | 1 | 1 | −9 |
f3 → f2 | 2 | 2 | −12 |
f4 → f3 | 3 | 3 | 0 |
f5 → f4 | 4 | 4 | +12 |
f6 → f5 | 5 | 5 | +9 |
f7 → f6 | 6 | 6 | 0 |
f8 → f7 | 7 | 0 | 0 |
f9 → f8 | 8 | 1 | −9 |
f10 → f9 | 9 | 2 | −12 |
f11 → f10 | 10 | 3 | 0 |
f12 → f11 | 11 | 4 | +12 |
f13 → f12 | 12 | 5 | +9 |
f14 → f13 | 13 | 6 | 0 |
As with the tetrad effect, the irregularities in Δi c 3 centred on the quarter- and the three-quarter-full shell are associated with the ability of electrons to keep apart better if their m l values have the same sign rather than opposite. This eases the general increase in interelectronic repulsion in the first quarter of the series, where electrons have m l values of the same sign, relative to the second quarter, where they have both positive and negative values (Table 4); similarly in the third quarter over the fourth quarter.
Note that the pattern of variations exhibited by i 3 and i 4 holds for other processes involving the removal of an electron from an f shell (see below). Note also, with respect to the formation of 5-valent compounds by praseodymium, that this has the lowest fifth ionization energy of the lanthanides (Kramida et al. 2016).
Metallic compounds
Nature of apparently bivalent compounds
Plots of ΔG for inter-valency conversion and other bi-f → tri-f processes against (Bn − i 3) showing approximate zero lines required to bring the graphs together (see text)
Other systems in figure 6
The scope of Fig. 6 can be enlarged beyond salt-like/metallic distributions by using a classification that focuses on the 4f n configuration rather than on valency (Johnson 1977). Compounds or ions in which the lanthanide has the same 4f configuration as does the dipositive ion in an [Xe]4f n state are described as bi-f; those with the same 4f n configuration as the tripositive ion are called tri-f. Figure 6 then describes the distribution of systems between di-f and tri-f states. This allows us to include the free, gaseous dipositive ions which have either [Xe]4f n or [Xe]4f n−15d ground states, and the metals which may be (M2+2e−) or (M3+3e−). The bonding contribution of the outer 5d electron in the tri-f systems varies between zero in the gaseous ions to substantial in metallic compounds such as CeS where it populates a conduction band. Intermediate situations such as the dipositive ions in aqueous solution or fluoride host lattices also appear in Fig. 6. In the next section we encounter a case in which the d electron in tri-f compounds participates in π-bonding with ligands. This too has been included in Fig. 6.
Organometallic compounds
The lanthanides form a number of organometallic compounds (see, e.g. Cotton and Wilkinson 1988: 968–973). These are generally trivalent, e.g. the cyclopentadienyl compounds Ln(η-C5H5)3. Compounds with other valencies are formed by elements for which these valencies are most stable, e.g. Ce(η-C5Η5)4, Sm(η-C5Η5)2, Eu(η-C5H5)2, and Yb(η-C5H5)2.
Of interest are the ions [Ln(η-C5H4R)3]− [R = Si(CH3)3] (Hitchcock et al. 2008; Fieser et al. 2015; Evans 2016). These exhibit two valencies. Most have the same structure and size as Ln(η-C5H4R)3, but those for samarium, europium, thulium and ytterbium are significantly larger. These are elements forming bivalent compounds (Table 1). Fieser et al. (2015) and Evans (2016) suggest that the lanthanide atoms in the latter ions are bivalent and have the configuration f n (M3+ = f n−1) while those in the former have the configuration f n−1d1 and are trivalent, the d electron being involved in the bonding. This pattern of valencies can be explained in a similar way to the pattern in metallic compounds; we have accordingly included these ions in Fig. 6.
Also of interest are the arene complexes [Ln(η-C6H3Bu 3 t )2]. Anderson et al. (1989) account for the relative stabilities and magnetic moments of these by proposing an f n−1d1s2 valence state for the lanthanide atoms. Promotion energies from f n s2 into this state vary in a similar way to i 3 and i 4 (above).
Footnotes
- 1.
We take the lanthanides (Ln) to be the elements from lanthanum to lutetium inclusive. We take lutetium to be a transition element and lanthanum to ytterbium “inner” transition elements.
- 2.
Although Slater’s theory fails for some systems, it holds for electrons outside cores (Johnson and Nelson 2014).
- 3.
This is one of several sets of integrals in the literature. It was proposed by Racah (1949). The superscripts are labels.
- 4.
Details and a more exact treatment will be given in Johnson and Nelson (paper in preparation).
- 5.
For an approximate treatment of this term, see Johnson (1982b).
- 6.
Taking ΔH θ ≈ ΔE.
- 7.
Values calculated from the standard enthalpies of formation of Ln3+(g) and of Ln3+(aq) relative to H+(aq) (Cordfunke and Konings 2001; Konings and Benes 2010; Konings et al. 2014; Kramida et al. 2016; Johnson and Nelson 2017a, b), and from the standard intrinsic enthalpy of formation of H+(aq). For the latter, we took the value recommended by Hünenberger and Reif (2011: 460), made consistent with the standard states used by NBS/NIST (Wagman et al. 1982) and JANAF (Chase 1998). The uncertainty in this value is considerable (at least ± 30 kJ mol−1), but this does not affect the precision of the relative values of h 3, most of which have uncertainties less than ±15 kJ mol−1. We have used absolute values to simplify the equations.
- 8.
Notes
Acknowledgements
We are grateful to a reviewer for helpful comments.
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