Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section H / 33 / b

A shell with angular momentum $l$ holds $2(2l+1)$ electrons. Successively filling the shells listed in the source gives the closed-shell totalsFor $\mathrm{A}{\mathrm{l}}_4\mathrm{C}{\mathrm{p}}_4^{\ast }$ the delocalized count is $4(3)-4=8$, matching $1S^{2}1P^6$. The tetrahedral aluminium core is consequently consistent with the jellium model. Counting the same eight electrons as though the tetrahedron were an ordinary closo Wade cluster would fail: a four-vertex closo framework would require $2n+2=10$ skeletal electrons. The competing models use different physical assumptions and should not be interchanged just because both involve a polyhedron.

In a skeletal description, each of the four naked Sn atoms retains an outward lone pair and contributes two electrons to the framework. Each aryl-substituted Ge atom contributes three electrons after assigning one to its Ge–aryl bond; it is not assigned an additional spectator lone pair in this counting. ThusThe PSEP prediction is a six-vertex closo octahedron, with the two protected Ge vertices and four naked Sn vertices; distortion does not change the topological classification. The alternative gross delocalized count $6(4)-2=22$ is not a listed jellium shell closure. Subtracting a lone pair from every atom would incorrectly remove electrons from the substituted Ge framework and give the wrong classification.

The central silicon contributes four electrons to the delocalized metal-core count. Six monovalent Cp* ligands each withdraw one electron, givingThis is a jellium closed shell, through the filled 2P shell. The ${\mu }_8$ label identifies an encapsulated silicon with eight aluminium contacts; it should not be counted as an ordinary exterior Wade vertex with an outward lone pair. The gross electron count rationalizes shell closure but does not alone reconstruct every Al–Al bond or all ligand positions.

The revised question page explicitly gives the cluster as $\{\mathrm{G}{\mathrm{a}}_{22}[\mathrm{N}(\mathrm{SiM}{\mathrm{e}}_3{)}_2{]}_{10}{\}}^{2-}$. Its charge supplies two electrons, soa jellium shell closure, through 1G. The reported structure has a Ga atom inside an eleven-atom Ga shell, with ten further ligand-bearing Ga units outside that core.^[ref]

The canonical exam copy omits the 2− charge. Taken literally as a neutral formula it gives 56 electrons, which is not a listed shell closure; assigning 58 without identifying this difference would be unjustified. The revised public source resolves the discrepancy rather than requiring an invented charge.^[ref]

For the neutral aluminium cluster,This closes the listed sequence through 1I and supports the jellium description of a large metalloid cluster. Cp* ligands are counted as monovalent electron-withdrawing surface groups; counting each as a five-electron donor to the entire delocalized core would mix a transition-metal local electron-counting convention with the present shell model. Electron closure is a useful stability criterion, not by itself proof of a perfectly spherical nuclear geometry.