Atomic Structure and Quantum Numbers: Study Notes and Practice Problems

Atomic Structure

Electromagnetic Radiation and Atomic Structure

The atomic structure of elements is closely related to the behavior of electrons, which are described by quantum mechanics and electromagnetic theory. Understanding the relationship between frequency, wavelength, and energy is essential for describing atomic transitions and electron configurations.

• Wavelength and Frequency: Electromagnetic waves have a wavelength (λ) and frequency (ν) related by the speed of light (c).

• Energy of Photons: The energy (E) of a photon is proportional to its frequency.

Key Equations:

• Relationship between wavelength and frequency: $c = \lambda \nu$

• Energy of a photon: $E = h \nu$ where h is Planck's constant ($6.626 \times 10^{-34}$ J·s).

Example: To find the wavelength of a radio signal with a frequency of 100.7 MHz, use $\lambda = \frac{c}{\nu}$.

Atomic Transitions and Emission

Electrons in atoms occupy discrete energy levels. When an electron transitions between levels, it emits or absorbs a photon whose energy equals the difference between the levels.

• Longest Wavelength Emission: Occurs when the energy difference between levels is smallest (e.g., $n=2 \rightarrow n=1$ in hydrogen).

• Energy Levels in Hydrogen: $E_n = -\frac{2.18 \times 10^{-18} \text{ J}}{n^2}$

Example: Identifying which transition emits the longest wavelength photon in hydrogen.

Quantum Numbers and Electron Configuration

Principal, Angular, Magnetic, and Spin Quantum Numbers

Quantum numbers describe the properties and locations of electrons in atoms:

• Principal Quantum Number (n): Indicates the energy level (shell).

• Angular Momentum Quantum Number (l): Indicates the subshell (shape of orbital). Possible values: $l = 0$ to $n-1$.

• Magnetic Quantum Number (m_l): Indicates the orientation of the orbital. Possible values: $m_l = -l$ to $+l$.

• Spin Quantum Number (m_s): Indicates the spin of the electron. Possible values: $m_s = +\frac{1}{2}$ or $-\frac{1}{2}$.

Example: For $n=3$, possible $l$ values are 0, 1, 2. For $l=2$, possible $m_l$ values are -2, -1, 0, 1, 2.

Subshells and Orbitals

Each subshell is defined by a specific value of $l$ and contains a set number of orbitals:

Subshell	l	Number of Orbitals
s	0	1
p	1	3
d	2	5
f	3	7

Electrons per Subshell: Each orbital can hold 2 electrons (one with $m_s = +\frac{1}{2}$, one with $m_s = -\frac{1}{2}$).

• s subshell: 2 electrons

• p subshell: 6 electrons

• d subshell: 10 electrons

• f subshell: 14 electrons

Electron Capacity of Energy Levels

The total number of orbitals in a principal energy level $n$ is $n^2$. The total number of electrons is $2n^2$.

• For $n=3$: $3^2 = 9$ orbitals, $2 \times 9 = 18$ electrons

• For $n=2$: $2^2 = 4$ orbitals, $2 \times 4 = 8$ electrons

• For $n=1$: $1^2 = 1$ orbital, $2 \times 1 = 2$ electrons

Valid Quantum Number Sets

Not all combinations of quantum numbers are valid. For a given $n$, $l$ must be less than $n$, and $m_l$ must be between $-l$ and $+l$.

• Example: $n=2, l=1, m_l=0, m_s=+\frac{1}{2}$ is valid.

• Example: $n=2, l=2$ is invalid because $l$ cannot equal $n$.

Summary Table: Quantum Numbers and Orbitals

Quantum Number	Symbol	Possible Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level (shell)
Angular Momentum	l	0 to n-1	Subshell (orbital shape)
Magnetic	m_l	-l to +l	Orbital orientation
Spin	m_s	+1/2, -1/2	Electron spin direction

Practice Problems (from worksheet)

• Calculate the wavelength of a radio signal in meters with a frequency of 100.7 MHz.

• Find the energy of a photon of visible light with a wavelength of 615 nm.

• Determine the frequency of radiation required to supply a given energy to a number of photons.

• Identify which electron transition in hydrogen emits the longest wavelength photon.

• List possible quantum numbers for given principal quantum numbers.

• Determine the number of electrons that can occupy a set of quantum numbers.

• Specify quantum numbers for subshells and calculate electron capacity.

• Identify valid sets of quantum numbers for electrons in atoms.

• Calculate the number of orbitals and electrons in each principal energy level.

Additional info: These notes expand upon the worksheet questions by providing definitions, equations, and context for quantum numbers, atomic transitions, and electron configurations.