Quantum-Mechanical Model of the Atom: Study Notes

Chapter 7: Quantum-Mechanical Model of the Atom

Overview

This chapter introduces the quantum-mechanical model of the atom, focusing on the wave and particle nature of light, atomic emission, quantum numbers, and the shapes and properties of atomic orbitals. These concepts are foundational for understanding atomic structure and chemical behavior.

The Wave Nature of Light

Electromagnetic Radiation

• Light is a form of electromagnetic radiation, consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation.

• Electromagnetic waves are characterized by their wavelength ($\lambda$), frequency ($\nu$), and amplitude (A).

• Wavelength ($\lambda$): Distance light travels in one cycle (meters, nanometers).

• Frequency ($\nu$): Number of cycles per second (Hz).

• Amplitude: Height from the center of the wave; relates to intensity.

• Relationship: $c = \lambda \nu$ where $c$ is the speed of light ($3.00 \times 10^8$ m/s).

Example: Green light with $\lambda = 515$ nm has $\nu = \frac{c}{\lambda} = 5.83 \times 10^{14}$ s$^{-1}$.

The Electromagnetic Spectrum

Continuous Spectrum

• The electromagnetic spectrum includes all possible wavelengths and frequencies of electromagnetic radiation.

• Most types of radiation are invisible to the human eye.

• Wavelength and frequency are inversely related: $\nu = \frac{c}{\lambda}$.

Example: FM radio at $\nu = 100.2$ MHz has $\lambda = 2.994$ m.

The Particle Nature of Light

Photoelectric Effect

• The photoelectric effect demonstrates that light has both particle and wave nature.

• Light energy is quantized in discrete packets called photons.

• Planck's Law: $E = h\nu = \frac{hc}{\lambda}$, where $h = 6.626 \times 10^{-34}$ J·s.

Example: Number of photons in a laser pulse: $N = \frac{E_{\text{total}}}{E_{\text{photon}}}$.

Atomic Emission and Spectroscopy

Historical Observations

• Elements emit characteristic colors when burned due to emission of light at specific wavelengths.

• Light emitted by atoms contains only certain distinct wavelengths unique to each element.

Atomic Spectroscopy

• Light from an element can be separated into individual wavelengths using a prism, producing an emission spectrum.

• Emission spectra consist of discrete lines rather than a continuous band.

The Bohr Model of the Atom

Key Features

• Electrons move in circular orbits around the nucleus.

• Each orbit has a fixed set of allowed energies.

• Electrons can transition between orbits, emitting or absorbing light.

Energy of Orbits

• Energy of an electron in an orbit: $E_n = -2.18 \times 10^{-18} \left(\frac{Z^2}{n^2}\right)$ J

• Energy difference between two orbits: $\Delta E = -2.18 \times 10^{-18} Z^2 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$

Example: Wavelength of light absorbed/emitted during electron transitions can be calculated using $E = h\nu = \frac{hc}{\lambda}$.

Limitations of the Bohr Model

• Cannot explain emission spectra from multi-electron atoms.

• No theoretical basis for quantized orbits.

• Served as a bridge to the development of quantum mechanics.

The de Broglie Wavelength

• All matter exhibits both wave and particle properties.

• de Broglie wavelength: $\lambda = \frac{h}{mv}$

• Significant for very small and fast-moving objects (e.g., electrons).

Example: Calculate $\lambda$ for a baseball and compare to an electron.

The Heisenberg Uncertainty Principle

• Position ($x$) and momentum ($p$) of a particle cannot both be known precisely.

• Uncertainty Principle: $\Delta x \cdot m \Delta v \geq \frac{h}{4\pi}$

• Quantum mechanics describes electrons as probabilistic, not deterministic.

Quantization in Quantum Mechanics

• Atomic energy levels are quantized: only specific, discrete values are allowed.

• Contrast with classical systems, which have continuous energy values.

Quantum Mechanics and Atoms

The Schrödinger Equation

• Relates the wave properties of electrons to their quantized energies.

• General form: $\hat{H}\Psi = E\Psi$

• $\Psi$ (wavefunction) describes the orbital shape, orientation, and size.

Quantum Numbers and Atomic Orbitals

Quantum Numbers

• Principal quantum number ($n$): Overall size and energy of orbital ($n = 1, 2, 3, ...$).

• Angular momentum quantum number ($l$): Shape of orbital ($l = 0, 1, ..., n-1$).

• Magnetic quantum number ($m_l$): Orientation of orbital ($m_l = -l, ..., +l$).

Value of $l$	Letter Designation
0	s
1	p
2	d
3	f

Value of $l$	Values of $m_l$
0	0
1	-1, 0, 1
2	-2, -1, 0, 1, 2
3	-3, -2, -1, 0, 1, 2, 3

Shells and Subshells

• Number of subshells in a shell: n

• Number of orbitals in any shell: $n^2$

• Number of orbitals in a subshell: $2l+1$

From Wavefunctions to Atomic Orbitals

• Orbitals describe the probability ($\Psi^2$) of finding an electron in a region of space.

• Orbitals are mathematical functions, not physical objects.

• Boundary surfaces enclose the volume where an electron is found most of the time (typically 95%).

Shapes of Atomic Orbitals

s Orbitals ($l = 0$)

• Spherically symmetric; lowest energy orbital is 1s.

• Probability density is highest at the nucleus and decreases with distance.

Radial Probability Distributions

• Show probability of finding an electron at a given radius from the nucleus.

• Radial nodes are regions where probability is zero; for s-orbitals, number of radial nodes = $n - l - 1$.

p Orbitals ($l = 1$)

• Each shell with $n \geq 2$ contains three p orbitals ($m_l = -1, 0, +1$).

• p orbitals have two lobes of electron density separated by one angular node.

Phases of Atomic Orbitals

• Orbital phases (signs) are important in bonding; they determine constructive or destructive interference.

*Additional info: These notes expand on the provided slides with definitions, formulas, and examples for clarity and completeness. Tables have been recreated for quantum numbers and orbital designations. The content is suitable for exam preparation in a General Chemistry college course, specifically Chapter 7: Quantum-Mechanical Model of the Atom.*