Quantum Numbers & Subshells — Quick Reference

Subshells and orbital counts

  • Subshells: s, p, d, f

  • Orbital counts per subshell: s→1 orbital,p→3,d→5,f→7s\to 1\,\text{orbital},\quad p\to 3,\quad d\to 5,\quad f\to 7

  • Each orbital holds 2 electrons due to spin degeneracy

Quantum numbers: n, l, ml, ms

  • Principal quantum number: n=1,2,3,…n=1,2,3,…

  • Angular momentum quantum number: l∈0,1,…,n−1l\in{0,1,\dots, n-1}

  • Magnetic quantum number: ml∈−l,−l+1,…,lm_l\in{-l,-l+1,\dots, l}

  • Spin quantum number: ms∈+12,−12m_s\in{+\tfrac{1}{2}, -\tfrac{1}{2}}

  • Mapping: l=0→s, l=1→p, l=2→d, l=3→fl=0\to s,\ l=1\to p,\ l=2\to d,\ l=3\to f

  • For example, if n=3  ⇒  l=0,1,2n=3\;\Rightarrow\; l=0,1,2 (s, p, d)

  • For each ll, possible m<em>lm<em>l values: if l=2⇒m</em>l∈−2,−1,0,1,2l=2\Rightarrow m</em>l\in{-2,-1,0,1,2}; if l=0⇒m<em>l=0l=0\Rightarrow m<em>l=0; if l=3⇒m</em>l∈−3,−2,−1,0,1,2,3l=3\Rightarrow m</em>l\in{-3,-2,-1,0,1,2,3}

Orbitals per shell and labeling

  • The letter denotes ll; the number denotes nn

  • For a given nn, the allowed subshells have: l=0,1,…,n−1l=0,1,\dots, n-1

  • In general: m_l values range from −l-l to +l+l (step 1)

  • Example: for f-subshell, l=3⇒ml∈−3,−2,−1,0,1,2,3l=3\Rightarrow m_l\in{-3,-2,-1,0,1,2,3}; 4 is not allowed

Electron capacity and counting

  • Number of orbitals in shell nn: n2n^2

  • Maximum electrons in shell nn: 2n22n^2

  • Example: for n=3n=3, orbitals = 99, electrons = 1818

Transitions and energy-wavelength relationship

  • Longer wavelength corresponds to lower energy transitions: larger λ\lambda implies smaller ΔE\Delta E

  • Energy of a photon from a transition: ΔE=hc/λ\Delta E = h c / \lambda

  • Conversely: λ=hc/ΔE\lambda = h c / \Delta E

  • Practical note: transitions are between discrete levels (not continuous), with allowed integer changes in quantum numbers