Past exam of the chemistry course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / 6 / a

For the outer $ns,np$ valence electrons, Slater's rules predict a modest initial rise down group 15, followed by approximately constant values. Write the effective nuclear charge as $Z_{\mathrm{eff}}=Z-S$, where $S$ is the shielding constant. Each atom has $n{s}^{2}n{p}^3$, so the other four electrons in the outer group contribute $4(0.35)=1.40$. Electrons in shell $n-1$ contribute $0.85$ each, and those deeper contribute $1.00$ each.^[ref]

Applying those weights gives the following original calculations:The increasing nuclear electric charge is largely offset by additional inner-electron shielding; once the preceding shell contains $18$ electrons, these simple weights give a plateau. The $n-1$ shell includes its d electrons. This trend concerns outer-shell values, not every orbital: a $1s$ electron is shielded only by its partner, so $Z_{\mathrm{eff}}(1s)=Z-0.30$ rises strongly down the group. The larger principal shell also means nearly constant outer $Z_{\mathrm{eff}}$ does not imply nearly constant atomic size or binding energy.

Photoelectron spectroscopy measures orbital-dependent binding energies from which an effective charge can be inferred using a stated model. Ultraviolet photoelectron spectroscopy accesses outer-shell ionizations; X-ray photoelectron spectroscopy accesses deeper core states. For an isolated gas-phase atom and a calibrated threshold, measure emitted-electron kinetic energy and use $I=h\nu -E_{\mathrm{kin}}$, with the appropriate energy reference. Optical atomic term spectra and series limits also constrain valence ionization energies.

A hydrogenic inference would set $I\simeq R_{\infty }{Z}_{\mathrm{eff}}^2/n_{\mathrm{eff}}^2$, hence $Z_{\mathrm{eff}}\simeq n_{\mathrm{eff}}\sqrt{I/R_{\infty }}$, with $R_{\infty }$ here an energy of approximately $13.6\mathrm{eV}$ and $n_{\mathrm{eff}}$ a chosen effective shell parameter. The measured binding energies are experimental; the extracted $Z_{\mathrm{eff}}$ values remain model dependent. They need not equal Slater screening parameters fitted to radial functions.

Unlike the equal $ns,np$ values in Slater's rules, spectroscopic values distinguish s from p because their penetration and shielding differ; the more tightly bound s states usually imply larger binding-energy-derived effective charges at a common shell parameter. Subshell, atomic-term and, especially for heavier atoms, spin-orbit coupling structure give further distinctions. Electron relaxation following ionization and electron correlation also matter. Thus real binding energies vary across the heavier group-15 atoms even where the simple Slater outer-shell estimates plateau; no universal constant correction to the Slater values follows.^[ref]

Phosphorus has electron configuration $1s^{2}2s^{2}2p^{6}3s^{2}3p^3$, with $Z=15$. For a $1s$ electron, the only shielding contribution in Slater's rules is $0.30$ from the other $1s$ electron:For a $3p$ electron, the other four electrons in the $3s,3p$ group contribute $4(0.35)$, the eight $n=2$ electrons contribute $8(0.85)$, and the two $1s$ electrons contribute $2(1.00)$:The compact core electron experiences nearly the full nuclear charge, while the valence electron is strongly shielded by the core. The two numbers describe different orbitals of the same atom. Outer electrons do not shield the core in this Slater approximation, and the selected electron never counts toward its own shielding.

Allen electronegativity is proportional to the average binding energy of the free ground-state atom's valence electrons. For an s,p main-group atom, use positive, spectroscopically inferred one-electron binding energies ${\epsilon }_s,{\epsilon }_p$ and their occupancies $n_s,n_p$:Here $C>0$ fixes the conventional scale; the one-electron energies are obtained from appropriate atomic ionization/term energies, not merely from the first ionization energy. For group 15 the weighted average is $(2{\epsilon }_s+3{\epsilon }_p)/5$.^[ref]

The approximate connection to effective nuclear charge is ${\epsilon }_{n\ell }\sim R_{\infty }[Z_{\mathrm{eff}}(n\ell )/n_{\mathrm{eff}}{]}^2$. At a fixed shell, stronger effective nuclear attraction generally raises the valence binding energies and therefore Allen electronegativity. Down a group, increasing shell size and changes in penetration offset that attraction. There is no universal relation ${\chi }_{\mathrm{Allen}}\propto Z_{\mathrm{eff}}$ across different shells, nor can one substitute a core-electron $Z_{\mathrm{eff}}$ for the valence average.