Chapter 7

1. Introduction to Quantum Theory

• Classical physics failed to explain phenomena at the atomic and subatomic level (e.g., blackbody radiation, photoelectric effect, atomic spectra).

• Quantum theory describes matter and energy at the atomic and subatomic level, proposing that energy exists in discrete packets called quanta.

2. Electromagnetic Radiation

• Electromagnetic radiation (EMR) consists of oscillating electric and magnetic fields that travel at the speed of light in a vacuum.

  • Wavelength ($\lambda$): Distance between two consecutive peaks or troughs of a wave (units: m, nm).

  • Frequency ($\nu$): Number of waves passing a point per unit time (units: Hz or $s^{-1}$).

  • Amplitude: Height of the wave from the origin to a crest.

  • Speed of Light ($c$): $3.00 \times 10^8 \text{ m/s}$ in a vacuum.

• Relationship between $c$, $\lambda$, and $\nu$: $c = \lambda \nu$

• Energy of a Photon ($E$):

  • Planck's equation: $E = h \nu$

  • Where $h$ is Planck's constant (6.626 \times 10^{-34} \text{ J\cdot s}).

  • Combining equations: $E = \frac{hc}{\lambda}$

2.1. Photoelectric Effect

• Emission of electrons from a metal surface when light shines on it.

• Explained by Einstein using Planck's quantum theory: light behaves as particles (photons) with energy $h\nu$.

• A minimum frequency (threshold frequency, ${\nu}_{0}$) is required to eject an electron, regardless of light intensity.

3. Atomic Spectra

• Continuous spectrum: Produced when white light passes through a prism, showing all wavelengths.

• Line spectrum: Specific wavelengths of light emitted or absorbed by excited atoms.

  • Emission spectrum: Light emitted by excited atoms, unique to each element.

  • Absorption spectrum: Dark lines appear in a continuous spectrum where specific wavelengths are absorbed by atoms.

• Bohr Model (Limitations):

  • Proposed electrons orbit the nucleus in fixed energy levels (quantized).

  • Electrons can transition between

  • energy levels by absorbing or emitting photons of specific energies.

  • Formula for energy levels in a hydrogen atom: $E<em>n = -R</em>H \left( \frac{1}{n^2} \right)$, where $R_H$ is the Rydberg constant and $n$ is the principal quantum number.

  • Failed for multi-electron atoms and did not explain fine structure of spectral lines.

4. Quantum Mechanics

4.1. Wave-Particle Duality

• De Broglie Wavelength: Proposed that particles (like electrons) can also exhibit wave-like properties.

  • $\lambda = \frac{h}{mv}$, where $m$ is mass and $v$ is velocity.

4.2. Heisenberg Uncertainty Principle

• It is impossible to simultaneously know precisely both the position ($\Delta x$) and momentum ($\Delta p$) of a particle.

  • $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$

4.3. Schrödinger Equation (Conceptual)

• A mathematical equation that describes the wave function ($\Psi$) of an electron in an atom.

• Solutions to the Schrödinger equation yield atomic orbitals, which represent probability distributions of finding an electron in space.

• $\Psi^2$ (probability density) gives the probability of finding an electron at a particular point in space.

5. Quantum Numbers

• Four quantum numbers describe the state of an electron in an atom:

  1. Principal Quantum Number ($n$):

     • Values: $1, 2, 3, …$

     • Describes the electron's main energy level (shell) and average distance from the nucleus.

     • Higher $n$ means higher energy and larger orbital size.

  2. Azimuthal (Angular Momentum) Quantum Number ($l$):

     • Values: $0, 1, 2, …, n-1$

     • Describes the shape of the orbital (subshell).

     • $l=0$: s orbital (spherical)

     • $l=1$: p orbital (dumbbell)

     • $l=2$: d orbital (more complex)

     • $l=3$: f orbital (even more complex)

  3. Magnetic Quantum Number ($m_l$):

     • Values: $-l, …, 0, …, +l$

     • Describes the orientation of the orbital in space.

     • For $l=0$ (s), $m_l = 0$ (1 orbital).

     • For $l=1$ (p), $m_l = -1, 0, +1$ (3 orbitals).

     • For $l=2$ (d), $m_l = -2, -1, 0, +1, +2$ (5 orbitals).

  4. Spin Quantum Number ($m_s$):

     • Values: $+\frac{1}{2}$ or $\text{-}\frac{1}{2}$

     • Describes the intrinsic angular momentum (spin) of an electron, either "spin up" or "spin down".

6. Atomic Orbitals and Their Shapes

• s orbitals ($l=0$): Spherical shape, increasing size with $n$ (e.g., 1s, 2s, 3s).

• p orbitals ($l=1$): Dumbbell shape, existing in three orientations ($p<em>x, p</em>y, p_z$) along the axes.

• d orbitals ($l=2$): More complex shapes, typically four cloverleaf shapes ($d<em>{xy}, d</em>{yz}, d<em>{xz}, d</em>{x^2-y^2}$) and one dumbbell with a toroid ($d_{z^2}$).

7. Electron Configurations

• The arrangement of electrons in an atom's orbitals.

• Governed by three main principles:

  1. Aufbau Principle: Electrons fill the lowest energy orbitals first (e.g., $1s \to 2s \to 2p$).

  2. Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers. Therefore, an atomic orbital can hold a maximum of two electrons, and these two electrons must have opposite spins.

  3. Hund's Rule: For degenerate orbitals (orbitals of the same energy), electrons will occupy separate orbitals with parallel spins before pairing up in any one orbital.

7.1. Orbital Diagrams

• Visual representation of electron configuration using boxes or lines for orbitals and arrows for electrons (up arrow for $+\frac{1}{2}$, down arrow for $\text{-}\frac{1}{2}$).

7.2. Notation

• Spectroscopic notation: e.g., $1s^2 2s^2 2p^6$ (superscript indicates number of electrons in that orbital).

• Noble gas notation: Shorthand using the preceding noble gas symbol in brackets (e.g., for Sodium (Na): $[Ne] 3s^1$).