(Oxygen's atomic number, cut off in the source text, is 8.) The nitrate ion NO₃⁻ has 1 nitrogen (7 electrons) + 3 oxygens (3×8 = 24 electrons) = 31 electrons from the neutral atoms, plus 1 extra electron for the ion's overall negative charge = 32 electrons total.

Orbital angular momentum is given by L = √[l(l+1)]·(h/2π) — it depends only on the azimuthal quantum number l, not on n or m.

Isotopes of the same element share the same atomic number (same number of protons, by definition) but differ in mass number — which means the number of neutrons is exactly what's different between them.

Solving the (non-relativistic) Schrödinger wave equation for the hydrogen atom naturally yields the principal (n), azimuthal (l), and magnetic (m) quantum numbers. The spin quantum number (s) isn't a solution of this equation at all — it's an additional property introduced separately (rooted in relativistic quantum mechanics) to fully describe an electron.

Neutral Ni ([Ar]3d⁸4s²) forms Ni²⁺ by losing both 4s electrons, leaving [Ar]3d⁸. Filling 5 d-orbitals with 8 electrons under Hund's rule: all 5 orbitals get one electron each first, then 3 more electrons pair up in 3 of them — leaving 2 orbitals singly occupied, i.e. 2 unpaired electrons.