Quantum Numbers, Electron Configuration, and Photons: Study Notes for General Chemistry

Quantum Numbers and Electronic Structure

Four Quantum Numbers

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 of an electron. Allowed values: $n = 1, 2, 3, \ldots$

• Angular Momentum Quantum Number (l): Describes the shape of the orbital. Allowed values: $l = 0, 1, 2, \ldots, (n-1)$

• Magnetic Quantum Number (m_l): Specifies the orientation of the orbital in space. Allowed values: $m_l = -l, -(l-1), \ldots, 0, \ldots, +(l-1), +l$

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

Example: For $n=2$, possible $l$ values are 0 and 1. For $l=1$, $m_l$ can be -1, 0, or +1.

Valid Sets of Quantum Numbers

Each electron in an atom must have a unique set of quantum numbers. The sets must follow the rules above. For example:

• For $n=3$, $l=2$, $m_l=-2$, $m_s=+\frac{1}{2}$ is valid.

• For $n=2$, $l=1$, $m_l=0$, $m_s=-\frac{1}{2}$ is valid.

Invalid sets violate the allowed ranges (e.g., $l$ cannot be equal to or greater than $n$).

Electron Configuration

Electron Configuration for Technetium (Tc, Z=43)

Electron configuration describes the arrangement of electrons in an atom's orbitals. The order is determined by the Aufbau principle, Pauli exclusion principle, and Hund's rule.

• Technetium (Tc, Z=43): Electron configuration: $[Kr]\ 4d^5\ 5s^2$

Example: The configuration shows that Tc has filled 5s and half-filled 4d orbitals.

Embedded Electron Configuration of Ions

When atoms form ions, electrons are added or removed from the outermost shell first. For example, the $Ti^{2+}$ ion (Titanium, Z=22):

• Neutral Ti: $[Ar]\ 4s^2\ 3d^2$

• $Ti^{2+}$: Remove two electrons from 4s: $[Ar]\ 3d^2$

Photons and Electromagnetic Radiation

Wavelength and Frequency of Photons

The energy and wavelength of photons are related to their frequency by the following equations:

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

• Wavelength ($\lambda$): Distance between successive wave peaks (m).

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

Key Equations:

• $c = \lambda \nu$

• $E = h \nu$

• $E = \frac{hc}{\lambda}$

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

Example: For a photon with $\nu = 3.44 \times 10^{15}\ \text{Hz}$:

• $\lambda = \frac{c}{\nu}$

• $E = h \nu$

Wavelength of an Electron (de Broglie Wavelength)

Particles such as electrons have wave-like properties. The de Broglie wavelength is given by:

• $\lambda = \frac{h}{mv}$

Where $m$ is mass and $v$ is velocity.

Example: For $m = 9.11 \times 10^{-31}\ \text{kg}$ and $v = 2.56\ \text{m/s}$:

• $\lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 2.56}$

Atomic Transitions and Photon Emission

Energy Levels and Electron Transitions

Electrons in atoms occupy discrete energy levels. When an electron moves between levels, it emits or absorbs a photon:

• Absorption: Electron moves to a higher energy level (photon absorbed).

• Emission: Electron moves to a lower energy level (photon emitted).

Energy of transition:

• $E = -R_H \frac{Z^2}{n^2}$

Where $R_H$ is the Rydberg constant ($2.18 \times 10^{-18}\ \text{J}$), $Z$ is atomic number, $n$ is principal quantum number.

Wavelength of emitted/absorbed photon:

• $\Delta E = E_{final} - E_{initial}$

• $\lambda = \frac{hc}{|\Delta E|}$

Longest and Shortest Wavelengths from Atomic Transitions

The longest wavelength corresponds to the smallest energy difference (e.g., $n=3$ to $n=2$), and the shortest wavelength to the largest energy difference (e.g., $n=\infty$ to $n=2$).

Transition	Energy Difference ($\Delta E$)	Wavelength ($\lambda$)
Largest $\Delta n$ (e.g., $n=\infty \to n=2$)	Maximum	Shortest
Smallest $\Delta n$ (e.g., $n=3 \to n=2$)	Minimum	Longest

Example: For $He^+$, use $Z=2$ in the energy equations.

Summary Table: Quantum Numbers

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

Additional info: These notes expand on the original questions by providing definitions, equations, and examples for quantum numbers, electron configurations, photon energy/wavelength, and atomic transitions, as relevant to General Chemistry topics (Ch.6 - Electronic Structure of Atoms).