College Physics - Electronic Structure and Periodic Properties of Elements

Electromagnetic Radiation

Electromagnetic radiation refers to energy waves that include visible light along with other types of radiation like X-rays and microwaves. When atoms absorb energy, electrons within these atoms become excited and then release energy as electromagnetic radiation. Light can be represented as consisting of two orthogonal vectors: an electric wave and a magnetic wave. The properties of light can be characterized by its wavelength (BB) and frequency (number of cycles per second, denoted as (\nu)). The speed of light (
(c)) is a constant, approximately equal to 2.998 \times 10^{8} \text{ m/s}.

Relationship Between Wavelength and Frequency

The relationship between wavelength and frequency is defined by the equation:
\lambda \cdot \nu = c
This shows that as the wavelength increases, the frequency decreases and vice versa, establishing an inverse relationship.

Example Calculation of Frequency

For instance, the frequency of red light (BB = 685 nm) can be calculated as follows:

  1. Convert nanometers to meters:
    685 \text{ nm} = 6.85 \times 10^{-7} \text{ m}

  2. Calculate frequency using:
    \nu = \frac{c}{\lambda} = \frac{3.00 \times 10^{8} \text{ m/s}}{6.85 \times 10^{-7} \text{ m}} = 4.38 \times 10^{14} \text{ s}^{-1}

Quanta and the Photoelectric Effect

In 1900, Max Planck proposed that energy comes in tiny packets called quanta, with energy defined by the equation:
E = h\nu
where (h) is Planck's constant h = 6.626 \times 10^{-34} \text{ J s}. In the photoelectric effect, certain metals emit electrons when struck by light, provided the incident light's energy exceeds a certain threshold. Einstein elaborated on this phenomenon, demonstrating the correspondence between light waves and photons.

Photoelectric Effect Example

For a designed switch that requires 6.7 \times 10^{-19} \text{ J/atom} to eject electrons, we must determine if light of wavelength 540 nm will suffice:

  • Convert 540 nm to energy:
    E = \frac{hc}{\lambda}
    Calculating this shows it is insufficient to cause electron ejection.

Bohr Model of the Atom

Niels Bohr provided a structure for the hydrogen atom based on light spectra from excited atoms. He proposed that electrons exist in fixed orbits around the nucleus, with distinct energy levels. The emission spectra from hydrogen can only be observed at specific wavelengths as electrons jump between energy levels. The allowed energy levels are quantized, defined by quantum numbers (n = 1, 2, 3, … ).

Understanding Energy Transitions and Spectra

The transitions between these levels correspond to the emission of light in quantized energies. For example, the Balmer series represents transitions that end at (n=2), which lies within the visible spectrum.

Quantum Mechanics and Wave-Particle Duality

Particle-Wave Duality: Louis de Broglie theorized that all matter has wave-like properties, leading to the understanding that electrons can behave both as particles and waves. A crucial aspect of quantum mechanics is the uncertainty principle established by Heisenberg, which states that it is impossible to precisely determine both the position and momentum of a particle, particularly electrons.

Schrödinger’s Wave Function: Erwin Schrödinger introduced a mathematical description where electrons are defined by a wave function (\Psi). The square of the wave function's amplitude (\Psi^{2}) represents the probability density of finding an electron in space.

Quantum Numbers

Electrons are characterized by four quantum numbers describing their energy state and location within an atom:

  • Principal Quantum Number (n): Indicates the energy shell (1, 2, 3, …).

  • Angular Momentum Quantum Number (l): Defines the shape of the orbital (s, p, d, f).

  • Magnetic Quantum Number (m_{l}): Describes the orientation of the orbital in space.

  • Spin Quantum Number (m_{s}): Indicates the spin direction of the electron ((\pm 1/2)).

Electron Configurations

Electron configurations denote how electrons are distributed among the various atomic orbitals. The Aufbau principle establishes that electrons fill the lowest energy levels first. The general filling sequence of orbitals can be complex due to anomalies in transition metals.

Practice: Electron Configurations

To determine the electron configurations of various elements, consider the atomic number, which dictates the number of electrons:

  • For Nitrogen ((N)), with 7 electrons: 1s^{2} 2s^{2} 2p^{3}.

  • For ions, adjust electron counts based on the charges (e.g. (N^{3-} adds three more electrons resulting in 1s^2 2s^2 2p^6$$).

Periodic Trends

The arrangement of elements based on their atomic number reveals periodic properties such as atomic radii, ionization energy, and electron affinity:

  • Atomic Radii Increases Down a Group: Due to additional electron shells.

  • Ionization Energy Increases Across a Period: As nuclear charge increases, electrons are held more tightly.

  • Electron Affinity Variations: Nonmetals, especially halogens, typically exhibit more negative electron affinity values compared to metals.

By understanding these foundational concepts in electronic structure and periodic properties, students can build a robust framework to analyze the behavior of elements and their interactions according to quantum theory.