The center of an atom is the nucleus; electrons are moving very fast around it.
Schrödinger introduced the wave function, which is essentially a probability distribution function for locating an electron in three-dimensional space.
The wave function yields probabilities, not exact positions. It tells us where an electron is likely to be found.
Quantum numbers arise from this wave function and describe regions around the nucleus where electrons are likely to be found. These numbers are: n,\ l,\ m{l},\ m{s}.
The key quantum numbers in this discussion are n (principal), l (angular momentum), and ml (magnetic); ms (spin) is also defined but discussed as less central for the spatial picture.
These quantum numbers determine the regions around the nucleus where a given electron resides (the orbitals and subshells).
Principal Quantum Number (n): Energy Level and Shell
n describes the size and energy level of the electron’s region; it is also referred to as the energy level or shell.
n is a positive integer: n = 1, 2, 3, \dots
As n increases, the energy increases and the average distance of the electron from the nucleus increases.
The energy level corresponds roughly to a row on the periodic table (with caveats to be discussed).
Angular Momentum Quantum Number (l) and Subshells
For a given n, l can take integer values from 0 up to n-1:
If l=0\Rightarrow s\ subshell
If l=1\Rightarrow p\ subshell
If l=2\Rightarrow d\ subshell
If l=3\Rightarrow f\ subshell
The number of subshells within a given n equals n. The greater the n, the more space (subshells) you have, allowing more orbitals and electrons.
The symbol for the subshell is named after the letter associated with l: s, p, d, f.
Magnetic Quantum Number (m_l) and Orbitals
In each subshell, there are orbitals with different spatial orientations.
The magnetic quantum number m_l determines the orientation of the orbital within a subshell.
For a given l, m_l can take values from -l to +l, in integer steps:
Number of orbitals in a subshell = 2l+1.
Examples:
For l=0\ (s): m_l = 0\Rightarrow 1\mathrm{orbital}.
For l=1\ (p): m_l = -1, 0, 1\Rightarrow 3\mathrm{orbitals}.
For l=2\ (d): m_l = -2,-1,0,1,2\Rightarrow 5\mathrm{orbitals}.
Spin Quantum Number (m_s)
The spin quantum number m_s can be either +\tfrac{1}{2} or -\tfrac{1}{2}, corresponding to the two possible spin states.
Each orbital can hold up to two electrons with opposite spins (Pauli exclusion principle).
Degenerate Orbitals and Subshells
Degenerate orbitals are orbitals within the same subshell that have the same energy (e.g., the three p orbitals in a given shell).
The term ‘‘degenerate’’ means equal energy for those orbitals within the same subshell.
So in a given subshell like p (l=1), the three orbitals are degenerate with respect to energy.
Electron Configurations, Aufbau Principle, and Rules
Ground-state electron configuration is the arrangement of electrons with the lowest possible energy.
Aufbau principle: electrons fill from the lowest energy level upward (start low and build up).
Pauli exclusion principle: an orbital can hold at most two electrons, and these two electrons must have opposite spins.
Hund’s rule: when filling degenerate orbitals (same energy within a subshell), place one electron in each orbital before pairing two electrons in any orbital.
These rules together explain the arrangement of electrons in atoms and why atoms interact the way they do.
Worked Examples of Electron Configurations
Hydrogen (H): atomic number 1 → 1 electron.
Start at the lowest energy level: n=1,\ l=0 (1s subshell).
The first (and only) electron goes into the 1s orbital with spin up: 1s^1.
shorthand notation: 1s^1.
Helium (He): atomic number 2 → 2 electrons.
Fill 1s with two electrons of opposite spins: 1s^2.
shorthand: 1s^2.
Sodium (Na, Z=11): language of filling beyond helium.
After filling 1s and 2s/2p, the next electrons go into the 3s subshell.
Electron configuration: ext{Na}: 1s^2\ 2s^2\ 2p^6\ 3s^1.
Explanation: the first energy level (n=1) is full, the second energy level (n=2) is full (2s^2, 2p^6), and there is one electron in the third energy level (3s^1).
Chlorine (Cl, Z=17):
Start as for Na and fill up to 3p.
Electron configuration: ext{Cl}: 1s^2\ 2s^2\ 2p^6\ 3s^2\ 3p^5.
Reasoning: after filling 1s^2 and 2s^2 2p^6 (10 electrons), add 3s^2 (2 more, total 12), then fill 3p with 5 electrons (3p^5) to reach 17.
Ionic bond formation (NaCl):
Sodium tends toward a noble gas-like configuration by removing one electron to achieve a full outer shell; chlorine tends toward a noble gas-like configuration by gaining one electron.
Sodium transfers 1 electron to chlorine, yielding NaCl: Na becomes Na⁺, Cl becomes Cl⁻, and both attain closed outer shells which stabilizes the system.
This is the essence of an ionic bond; later we will discuss covalent bonds where electrons are shared.
Connection to the Periodic Table
The arrangement of electrons into shells and subshells explains why the periodic table is structured as it is: rows correspond to principal quantum levels and columns reflect filling patterns of subshells, with noble gases representing full outer shells.
As you move to higher n, more subshells become available, allowing more electrons and more complex chemistry.
4p Example and Quantum Number Values
For a four p configuration (4p):
The principal quantum number is n=4 (energy level of the outermost electrons).
The angular momentum quantum number is l=1 (p subshell).
The magnetic quantum number can take values m_l\in{-1,0,1} (three degenerate orbitals).
There are 2l+1=3 orbitals in the 4p subshell; each orbital can hold 2 electrons (paired or unpaired according to Hund’s rule).
Terminology and Significance
Ground state: the lowest-energy arrangement of electrons for an atom.
Degenerate orbitals: orbitals within the same subshell that have the same energy.
Aufbau principle: electrons fill from low to high energy levels to achieve the ground state.
Pauli exclusion principle: each orbital holds at most two electrons with opposite spins.
Hund’s rule: maximize unpaired electrons in degenerate orbitals before pairing.
Practical and Philosophical Relevance
The quantum-number-based model explains chemical bonding, bonding types (ionic vs covalent), and material properties.
The periodic table’s structure arises from the filling order of electrons in subshells, which reflects underlying quantum mechanical rules.
The language of quantum numbers, orbitals, and electron configurations connects microscopic electron behavior to macroscopic chemical behavior and material properties.
Course Context and Scheduling Note
The next 26 slides come from chapter five and are presented as not immediately on quizzes/exams; they will be revisited in about a month.
While not tested yet, the material provides essential context and foundation for later chapters and for understanding bonding and interactions.
The instructor emphasizes that the student should keep in mind how the chapter two material links to the ensuing discussion on quantum mechanics and bonding.
Summary of Key Concepts to Remember
Wave function describes probability distribution of an electron’s position in 3D space.
Quantum numbers: n,\ l,\ ml,\ ms determine electron regions around the nucleus.
Subshells named by l: s (l=0), p (l=1), d (l=2), f (l=3); numbers of orbitals per subshell: 2l+1\, (s:1, p:3, d:5, f:7).
Orbital capacity: each orbital holds up to two electrons with opposite spins (Pauli exclusion).
Degenerate orbitals share the same energy within a subshell.
Aufbau principle + Pauli exclusion + Hund’s rule govern electron configurations and chemical behavior.