Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 10

Ultrashort metal–metal contacts are favoured when the valence configuration fills strongly metal–metal-bonding orbitals while leaving their antibonding partners empty. Early or middle transition metals in low oxidation states can retain enough d electrons for multiple bonding without the unavoidable antibonding occupation of a more filled d shell. Good radial and angular overlap, a compatible metal–metal axis, and low competing ligand-field stabilization all matter; a short contact should be interpreted together with an orbital/electron-counting model, not automatically called a high-order bond.

Ligands should support a geometry in which the relevant d orbitals can overlap and protect the low-valent pair from oligomerization or oxidative attack. Bulky aryl-substituted bridging amidinates can provide this protection while maintaining an appropriate small coordination number. Strongly coordinating axial ligands or inappropriate pi interactions can compete with metal–metal bonding, especially its weaker delta components. For quadruple or quintuple bonds, retaining two appropriately oriented delta-capable orbitals is particularly important; a geometry optimized only for metal–ligand bonding can remove that advantage.

Ligand bridges need not literally squeeze two otherwise unbound metals together. Comparing bond lengths across oxidation states and across related ligands helps distinguish electronic multiple bonding from enforced steric compression. The lantern-type dichromium experiment in (ii) is informative because reduction changes the electron occupation within essentially the same three-bridge supporting framework.^[ref]

chromium is the characteristic assignment, giving Cr(I)–Cr(I) with a formal quintuple bond. Each formamidinate is monoanionic, so $2x-3=-1$ gives $x=+1$. chromium(I) is $d^5$ and the pair supplies ten d electrons. Relative to a metal–metal $z$ axis, the idealized bonding combinations are $d_{z^2}$ sigma, $d_{xz},d_{yz}$ pi and $d_{xy},d_{x^2-y^2}$ delta:End-on sigma overlap gives the largest splitting, pi overlap a smaller splitting, and delta overlap the weakest. A useful qualitative ordering places bonding sigma lowest, then the two pi combinations, then the two delta combinations; all are occupied. Their antibonding partners are empty, with the weakly split delta-star levels nearest their bonding counterparts. The actual three-bridge ligand field can mix and split symmetry-compatible levels, so the diagram is a bond-axis model rather than five perfectly degenerate metal d levels in the molecular complex.

The identified lantern-type anion has an experimentally determined Cr–Cr distance near $1.740$ Å; reduction of its nine-bonding-electron precursor supplies the tenth electron. This corroborates the model more specifically than citing “short bonds” in an unrelated framework.^[ref]

Use the balanced aqueous-acid reactionA Hess cycle first atomizes the metal, then ionizes its gas-phase atoms, then hydrates the triply charged gas ions:Close the reference leg by combining the three electrons with $3{\mathrm{H}}^{+}(\mathrm{aq})$ to make $\frac{3}{2}{\mathrm{H}}_2(\mathrm{g})$. Calling that element-independent contribution $C_H$ givesA single-ion standard enthalpy of formation uses the conventional hydrogen-ion reference; it must not be treated as a balanced creation of net charge from neutral metal without an electron/reference term. The constant hydrogen leg does not affect comparisons along the lanthanide series.

The large positive sum of ionization energies and large negative hydration enthalpy dominate, with a smaller but significant positive atomization term. Ionization depends on effective nuclear attraction, orbital energies and the stability of successive gas-ion electron configurations; removing the third electron may require disruption of a particularly stable divalent f configuration. Atomization depends on the metal's cohesive bonding and valence participation, and need not vary perfectly smoothly.

Hydration becomes more favourable for higher charge density. All the compared ions have charge $+3$, so smaller effective ionic radius generally gives more negative hydration enthalpy. Incomplete shielding by the 4f shell produces the lanthanide contraction and strengthens ion–water attraction along the series. A Born-type electrostatic estimate has a stabilization scale proportional to $-z^2/r$, but hydration-shell structure, water polarization and temperature dependence prevent treating that crude scale as an exact enthalpy formula. The hydration trend is smoother than the sharp configuration-dependent ionization anomalies. The measured formation enthalpy reflects cancellation of large terms, so comparison of just one term can give a misleading prediction.

The unusually stable divalent configurations Eu(II) $4f^7$ and Yb(II) $4f^{14}$ make their third ionization particularly costly. Forming Eu(III), $4f^6$, or Yb(III), $4f^{13}$ removes an electron from a half-filled or filled f shell. The positive ionization-energy contribution therefore increases relative to the smoother background across the series.

Their trivalent ionic radii and hydration enthalpies do not supply a matching abrupt extra stabilization. Differences in atomization and the divalent character of elemental Eu and Yb partly compensate the ionization anomaly but do not erase the net less favourable trivalent aqueous formation enthalpy. This explanation concerns formation of the $+3$ aqueous ions, not a claim that $\mathrm{E}{\mathrm{u}}^{3+}$ or $\mathrm{Y}{\mathrm{b}}^{3+}$ is intrinsically the most stable isolated f configuration.