Quantum Mechanics, Atomic Structure, and Electromagnetic Radiation: General Chemistry Study Notes

Quantum Mechanics and Atomic Structure

Quantum Numbers and Atomic Orbitals

Quantum numbers are used to describe the properties and locations of electrons in atoms. Each electron in an atom is characterized by a unique set of four quantum numbers:

• Principal quantum number (n): Indicates the main energy level or shell. Possible values: n = 1, 2, 3, ...

• Angular momentum quantum number (l): Defines the shape of the orbital. Possible values: l = 0 to n-1.

• Magnetic quantum number (ml): Specifies the orientation of the orbital. Possible values: ml = -l to +l.

• Spin quantum number (ms): Describes the spin of the electron. Possible values: ms = +1/2 or -1/2.

Example: For n = 4, possible values of l are 0, 1, 2, 3. For each l, ml ranges from -l to +l.

Orbitals in a Principal Level: The number of orbitals in a shell is given by n2. For n = 3, there are 9 orbitals.

Possible values for a 2p orbital: n = 2, l = 1, ml = -1, 0, +1.

Electron Configuration and Orbital Properties

• 4d Orbital: For n = 4, l = 2 (d orbital), ml = -2, -1, 0, +1, +2, ms = +1/2 or -1/2.

• Number of Orbitals in Fourth Shell: For n = 4, number of orbitals = 16.

Closest and Farthest Orbitals: Electrons in s orbitals (l = 0) are closest to the nucleus, while those in higher l orbitals (such as f, l = 3) are farthest.

Quantum Number Sets

Each electron is described by a unique set of quantum numbers. Invalid sets may have values outside allowed ranges (e.g., l >= n, ml outside -l to +l, or ms not ±1/2).

Electromagnetic Radiation and Photons

Wavelength, Frequency, and Energy Calculations

Electromagnetic radiation is characterized by its wavelength (λ), frequency (ν), and energy (E). The relationships are:

• Speed of light: $c = \lambda \nu$

• Energy of a photon: $E = h \nu = \dfrac{hc}{\lambda}$

• Planck's constant: $h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$

• Speed of light: $c = 3.00 \times 10^8 \text{ m/s}$

Example: To find the wavelength of red light with frequency $4.84 \times 10^{14}$ Hz:

• $\lambda = \dfrac{c}{\nu}$

• Substitute values to solve for λ in nm.

Energy of a Photon: For yellow light (λ = 575 nm):

• $E = \dfrac{hc}{\lambda}$

• Convert λ to meters before calculation.

Energy in Moles of Photons: Multiply energy per photon by Avogadro's number ($6.022 \times 10^{23}$) and number of moles.

Absorption and Emission in Hydrogen Atom

Electron transitions in hydrogen atoms correspond to absorption or emission of photons. The frequency (and energy) of the photon depends on the difference in energy levels:

• Absorption: Electron moves to a higher energy level (n increases).

• Emission: Electron moves to a lower energy level (n decreases).

• Smallest frequency photon: Corresponds to the smallest energy difference between levels.

Energy difference formula: $\Delta E = -R_H \left( \dfrac{1}{n_f^2} - \dfrac{1}{n_i^2} \right)$, where $R_H$ is the Rydberg constant.

Wave-Particle Duality

de Broglie Wavelength

The de Broglie wavelength describes the wave-like behavior of particles:

• $\lambda = \dfrac{h}{mv}$

• Where h is Planck's constant, m is mass, and v is velocity.

Example: For an electron with velocity $2.70 \times 10^6$ m/s and mass $9.11 \times 10^{-31}$ kg, substitute values to find λ.

Summary Table: Quantum Numbers and Orbitals

Quantum Number	Symbol	Allowed 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:

• Questions cover topics from Ch.8 (Quantum-Mechanical Model of the Atom), Ch.2 (Atoms & Elements), and Ch.3 (Molecules and Compounds), as well as mathematical operations relevant to quantum chemistry.

• Some questions require calculation using physical constants and conversion between units (e.g., nm to m).