Atomic Structure Summary

Electromagnetic Radiation

• Wavelength ($\lambda$): Distance between crests or troughs; S.I. unit is meter (m).
• Frequency ($f$): Number of complete wavelengths passing a point per second; unit is hertz (Hz) or s⁻¹.
• Inverse relationship between wavelength and frequency: $f = (1/\lambda) c$, where $c$ is the speed of light.

Quantum Theory

• Energy of a quantum: $E = hf$, where $h$ is Planck's constant ($6.63 x 10^{-34} Js$).

Bohr's Model of Hydrogen Atom

• Electrons move in circular orbits around the nucleus.
• Electrons remain in fixed orbits without emitting energy.
• Energy-level postulate: Electrons can only have specific energy values.
• Transitions between energy levels involve energy:
  • Higher to lower: energy emitted as a photon.
  • Ground state: all electrons in lowest possible energy levels.
  • Excited state: electron moved to a higher energy level.

Balmer Series

• Balmer's equation: $f = C (1/n<em>i^2 - 1/n</em>f^2)$, where $C = 3.29 x 10^{15} s^{-1}$, $n = 3, 4, 5$.
• Energy difference calculation: $E = -R<em>H (1/n</em>f^2)$, where $n<em>1$ and $n</em>2$ are integers and $n<em>2 > n</em>1$, $R_H = 2.179 x 10^{-18} J$.
• As $n$ increases, electron is completely separated from nucleus, energy approaches zero.

Energy Changes in Atoms

• Energy absorbed: electron moves to higher energy level.
• Energy emitted: electron moves to lower energy level, releases photon.
• Overall energy change: $\Delta E = E<em>f - E</em>i = R<em>H (1/n</em>i^2 - 1/n_f^2)$.
• Balmer's constant: $f = C [1/n<em>i^2 - 1/n</em>f^2]$, where $C = R_H/h = 3.29 x 10^{15} s^{-1}$.

De Broglie Principle

• Wavelength of electron: $\lambda = h/mv$ (De Broglie equation).
• Combining equations leads to $E = mc^2$.

Schrodinger's Equation

• Wave and particle-like behaviors of electron: $H\Psi = E\Psi$, where H is Hamiltonian operator, E is energy, and \Psi is wave function.
• $\Psi^2$ gives probability density of finding electron.

Orbital & Quantum Numbers

• Each electron described by four quantum numbers: $n$, $l$, $m<em>l$, $m</em>s$.
  • Principal Quantum Number ($n$): energy level; $n = 1, 2, 3,…$ (shells K, L, M, N,…).
  • Azimuthal Quantum Number ($l$): shape of orbital; $l = 0, 1, 2,…, (n-1)$ (s, p, d, f,…).
  • Magnetic Quantum Number ($m<em>l$): spatial orientation; $m</em>l = -l,…, 0,…, +l$.
  • Spin Quantum Number ($m<em>s$): electron spin; $m</em>s = \pm \frac{1}{2}$.
• Each atomic orbital can accommodate max of 2 electrons (Pauli exclusion principle).

Shape of Orbitals

• s orbital: spherically symmetrical; size increases with $n$ (1s < 2s < 3s).
• p orbitals: dumbbell shaped, node at nucleus; three components ($p<em>x$, $p</em>y$, $p_z$).
• d orbitals: five components with specific shapes.
• f orbitals: seven components, complex shapes.

Electron Configuration and Aufbau Principle

• Electron configuration: distribution of electrons among sub-orbitals.
• Aufbau principle: fill sub-orbitals in order of increasing energy (1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f).

Pauli Exclusion Principle

• No two electrons can have identical sets of four quantum numbers.

Hund's Rule

• Electrons fill separate orbitals with same spin before pairing.

Magnetic Properties

• Paramagnetic: substances with unpaired electrons, attracted to magnetic field.
• Diamagnetic: substances with paired electrons, repelled by magnetic field.