Electron Orbitals, Quantum Numbers, and Aufbau Principle

Quantum numbers, shells, and subshells

• The electron in an atom is described by four quantum numbers:

  • Principal quantum number: n\in {1,2,3,\dots}, which defines the shell (K for n=1, L for n=2, M for n=3, N for n=4, etc.).
  • The second quantum number is the angular momentum quantum number: l=0,1,2,\dots, n-1.
  • The third quantum number is the magnetic quantum number: m_l=-l,-l+1,\dots,+l.
  • The fourth quantum number is the spin quantum number: ms=+\tfrac{1}{2} or ms=-\tfrac{1}{2}.

• Relationship among the quantum numbers and orbitals:

  • For a given shell n, the possible values of l range from 0 to n-1.
  • For each l, there are 2l+1 magnetic orientations corresponding to the orbitals in that subshell (the degeneracy of the subshell).
  • Each orbital can hold up to two electrons with opposite spins (Pauli exclusion principle).
  • s, p, d, f notation corresponds to l=0,1,2,3 respectively (SPDF notation).

• Number of orbitals and maximum electrons:

  • In a subshell with angular momentum l there are 2l+1 orbitals.
  • Each orbital holds up to two electrons, so the subshell can hold 2(2l+1) electrons.
  • The maximum number of electrons in shell n is 2n^2 (e.g., 2 in the 1st shell, 8 in the 2nd shell, 18 in the 3rd shell, etc.).

• Key points about energy and degeneracy (from the transcript):

  • In a given shell, orbitals with higher l generally have higher energy (e.g., within the same shell, 2p is higher in energy than 2s; 3d higher than 3p, which is higher than 3s).
  • Orbitals within the same shell are degenerate in a hydrogen-like atom (same energy for 3s, 3p, 3d in the hydrogen case); in multi-electron atoms, shielding and electron–electron interactions split these energies.
  • The energy levels also depend on how far the shell is from the nucleus; energy generally increases with increasing n, but there are overlaps between shells (e.g., 4s can be filled before 3d).

Shells, subshells, and orbital shapes

• Shell naming (K, L, M, N, …) and corresponding n:

  • n=1\rightarrow K shell
  • n=2\rightarrow L shell
  • n=3\rightarrow M shell, etc.

• Subshells within shells:

  • n = 1: only l=0 (1s subshell)
  • n = 2: l=0,1 (2s and 2p subshells)
  • n = 3: l=0,1,2 (3s, 3p, 3d subshells)
  • n = 4: l=0,1,2,3 (4s, 4p, 4d, 4f subshells)

• Orbital shapes and labels:

  • s subshell: spherical shape; electron density highest at the center; l = 0; ml = 0 only.
  • p subshell: dumbbell-shaped with two lobes; l = 1; ml ∈ {−1, 0, +1}; three p orbitals oriented along the axes (x, y, z).
  • d subshell: five orbitals with four lobes around, plus a donut-like shape (often described as five d orbitals; ml ∈ {−2,−1,0,1,2}).
  • f subshell: seven orbitals; much more complex shapes (not detailed in the transcript and often omitted for introductory discussions).

• Degeneracy and filling in the same shell:

  • Within a given shell, orbitals of the same energy are described as degenerate (e.g., the three 2p orbitals have the same energy in a simple view, and the five 3d orbitals are degenerate in a single-shell picture).
  • The three 2p orbitals are oriented along different axes (x, y, z) and provide three possible spatial orientations for electrons in that subshell.

• Nodes and electron density:

  • The p orbitals have a nodal plane where the two lobes meet; the electron probability density is zero at the nucleus for a p orbital between the lobes.
  • The s orbital is spherical and has its highest probability inside the sphere around the nucleus; probability reduces as you move outward.

• Electron capacity per subshell (example):

  • 1s: 2 electrons
  • 2s: 2 electrons; 2p: 6 electrons (three 2p orbitals, each can hold 2 electrons)
  • 3s: 2, 3p: 6, 3d: 10 (five 3d orbitals, each can hold 2 electrons)

Orbitals, electrons, and electron configurations

• How electrons occupy orbitals ( Aufbau and Pauli rules):

  • Aufbau principle: electrons fill the lowest energy orbitals first, building up from inner shells to outer shells.
  • Electrons fill in order of increasing energy within the atom (lowest energy orbitals first).
  • The lowest energy orbitals are filled before moving to higher-energy orbitals, following the general energy trend determined by n and l.
  • In practice, this leads to the observed filling order such as 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, etc.
  • Overlaps of energy levels (e.g., 4s is filled before 3d) arise because of the way orbital energies shift with electron–electron interactions and shielding.
  • A mnemonic device is often used to memorize the order, but understanding the principles helps predict the order.

• Notation for electron configurations:

  • Subshell notation: use the principal quantum number n, the letter corresponding to l (s, p, d, f), and a superscript indicating the number of electrons in that subshell, e.g.,
  • Hydrogen: \text{1s}^1
  • Helium: \text{1s}^2
  • For example:
  • Helium: 1\text{s}^2
  • Lithium: 1\text{s}^2\ 2\text{s}^1
  • Beryllium: 1\text{s}^2\ 2\text{s}^2
  • Boron: 1\text{s}^2\ 2\text{s}^2\ 2\text{p}^1
  • Orbital diagrams (energy level diagrams) use horizontal lines for each orbital with arrows indicating electrons and their spins:
  • A line labeled for each orbital (e.g., 1s, 2s, 2p, etc.).
  • Arrows show spin, up for +1/2 and down for -1/2; paired electrons have opposite arrows.

• Example electron configurations and orbital diagrams (as discussed in the transcript):

  • Helium: 1s^2 ; orbital diagram shows one 1s orbital with two electrons of opposite spins (↑↓).
  • Lithium: 1s^2 2s^1 ; orbital diagram shows two electrons in 1s (paired) and one electron in 2s.
  • Beryllium: 1s^2 2s^2 ; orbital diagram shows two electrons in 1s (paired) and two in 2s (paired).
  • Boron: 1s^2 2s^2 2p^1 ; the fifth electron goes into one of the three 2p orbitals (degenerate in energy) with a single electron; by convention, it can occupy any of the 2p orbitals with a particular spin. This illustrates Hund’s rule idea without naming it explicitly in the transcript.

• Hund’s rule (implicit in the discussion):

  • When filling degenerate orbitals (same energy) within a subshell, electrons occupy different orbitals with parallel spins before pairing in the same orbital.
  • Example: In a 2p subshell with three degenerate orbitals, the first three electrons will singly occupy each of the 2p orbitals with parallel spins before any pairing occurs.

• Notions of valence and shielding (electrostatics):

  • Valence electrons: electrons in the outermost shell; they are most involved in bonding and chemical behavior.
  • Shielding/screening: inner (core) electrons reduce the full positive charge felt by outer electrons; outer electrons feel an effective nuclear charge less than the total nuclear charge.
  • Example from the transcript: If the nucleus has +11 charge (11 protons) and inner electrons sum to 10, the outermost electron experiences an effective nuclear charge of Z_{ ext{eff}} = Z - S = 11 - 10 = 1.
  • This shielding explains why outer-shell electrons are less strongly bound and why energy differences arise between levels (e.g., 2s vs 2p vs higher shells).

• Hydrogen-like vs multi-electron atoms:

  • For single-electron atoms or ions (e.g., hydrogen, He^+, Li^{2+}), electrons fill according to simple energy ordering with degeneracy within a shell (as described above).
  • In multi-electron atoms, electron–electron interactions and shielding cause splitting of energies between orbitals of the same principal quantum number (e.g., 2s vs 2p; 3s vs 3p vs 3d; 4s vs 3d, etc.).

Practical implications and quick references

• Orbital capacities and orientations (summary):

  • s: 1 orbital (ml = 0), capacity 2 electrons
  • p: 3 orbitals (ml = −1, 0, +1), capacity 6 electrons
  • d: 5 orbitals (ml = −2, −1, 0, +1, +2), capacity 10 electrons
  • f: 7 orbitals (ml = −3, −2, −1, 0, +1, +2, +3), capacity 14 electrons

• Key formulas in this topic (reiterated for study):

  • Angular momentum quantum number: l=0,1,2,…,n-1
  • Magnetic quantum number: m_l=-l,-l+1,…,+l
  • Spin quantum number: m_s=\pm \tfrac{1}{2}
  • Subshell electron capacity: 2(2l+1)
  • Shell capacity (max electrons): 2n^2

• Real-world relevance:

  • The arrangement of electrons determines chemical properties and periodic trends (valence electron configuration influences bonding, ionization energies, and sizes).
  • Shielding and effective nuclear charge help explain why atoms of the same group behave similarly and why the periodic table is structured the way it is.

• Reminders about the learning approach from the transcript:

  • Look for trends in the data (how energy changes with n and l).
  • Use the Aufbau principle to predict electron configurations.
  • Recognize that, in teaching, even if you memorize the filling order, understanding why the order appears (energy overlap, shielding) is crucial for deeper comprehension.

• Examples to practice (from transcript):

  • Hydrogen: electron goes to the lowest energy orbital: 1\text{s}^1
  • Helium: 1\text{s}^2
  • Lithium: 1\text{s}^2\ 2\text{s}^1
  • Beryllium: 1\text{s}^2\ 2\text{s}^2
  • Boron: 1\text{s}^2\ 2\text{s}^2\ 2\text{p}^1 (the fifth electron occupies one of the degenerate 2p orbitals)

• Note on notation and learning flow:

  • SPDF notation maps to subshells: s, p, d, f correspond to l=0,1,2,3.
  • The energy ordering and sharing of electrons among subshells is central to understanding the structure of atoms and their chemical behavior.