Quantum Mechanics and Electron Configuration: Atomic Orbitals and Quantum Numbers

Quantum Mechanics and Electron Configuration

Atomic Orbitals and Quantum Numbers

Quantum mechanics describes the behavior of electrons in atoms using mathematical functions called orbitals. Each electron in an atom is characterized by a set of quantum numbers that define its energy, shape, and orientation in space.

• Principal Quantum Number (n): Indicates the main energy level or shell of an electron. Values: n = 1, 2, 3, ...

• Angular Momentum Quantum Number (l): Defines the shape of the orbital. Values: l = 0 (s), 1 (p), 2 (d), 3 (f)

• Magnetic Quantum Number (ml): Specifies the orientation of the orbital in space. Values: ml = -l to +l

• Spin Quantum Number (ms): Describes the spin of the electron. Values: ms = +1/2 or -1/2

Example: For a 3p electron: n = 3, l = 1, ml = -1, 0, or +1, ms = +1/2 or -1/2

Energy Levels and Subshells

Electrons occupy energy levels (shells) and subshells (s, p, d, f) according to the Aufbau principle, which states that electrons fill the lowest energy orbitals first.

• Order of Filling: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s ...

• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers.

• Hund's Rule: Electrons fill degenerate orbitals singly before pairing.

Example: The electron configuration of oxygen (atomic number 8) is 1s2 2s2 2p4.

Electron Configuration Notation

Electron configurations are written using the notation: [energy level][subshell]number of electrons. For example, carbon: 1s2 2s2 2p2.

Mathematical Representation of Orbitals

Atomic orbitals are described by wave functions, which are solutions to the Schrödinger equation for the hydrogen atom:

• Schrödinger Equation:

$\hat{H}\Psi = E\Psi$

• Radial Probability Density: The probability of finding an electron at a certain distance from the nucleus is given by the square of the wave function.

Energy of Electron in Hydrogen Atom

The energy of an electron in a hydrogen atom is given by:

$E_n = -\frac{R_H}{n^2}$

where $R_H$ is the Rydberg constant ($2.18 \times 10^{-18}$ J), and $n$ is the principal quantum number.

Bohr Model and Energy Levels

The Bohr model describes electrons in fixed orbits around the nucleus, with quantized energy levels. The energy difference between levels determines the wavelength of light emitted or absorbed:

$\Delta E = E_{final} - E_{initial}$

$E = h\nu = \frac{hc}{\lambda}$

where $h$ is Planck's constant, $\nu$ is frequency, $c$ is the speed of light, and $\lambda$ is wavelength.

Summary Table: Quantum Numbers and Their Meaning

Quantum Number	Symbol	Possible Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level (shell)
Angular Momentum	l	0 to n-1	Orbital shape (s, p, d, f)
Magnetic	ml	-l to +l	Orbital orientation
Spin	ms	+1/2, -1/2	Electron spin direction

Additional info: Some context and equations have been inferred and expanded for completeness and clarity, based on standard General Chemistry curriculum.