Quantum Numbers, Atomic Models, and Nuclear Reactions in Physical Chemistry

When filling these orbitals, we follow specific rules to determine the electronic configuration. A common error is attempting to place electrons in non-existent subshells, such as a $2d$ orbital; in reality, the $d$-subshell only begins at $n=3$.

Atomic Models and Light-Matter Interaction

Before modern quantum mechanics, the Bohr model provided a crucial stepping stone by describing electrons in discrete, quantized orbits.

The Bohr Model

In the Bohr model, the energy of an electron in a hydrogen-like species (an atom with one electron, such as $\text{He}^+$ or $\text{Li}^{2+}$) is quantized. The energy levels are related to the atomic number $Z$ and the orbit number $n$. Key relationships include:

• The radius of the $n$-th orbit is proportional to $\frac{n^2}{Z}$.
• The energy of an electron in a specific orbit is proportional to $-\frac{Z^2}{n^2}$.

When an electron jumps from a higher energy level ($n_{high}$) to a lower one ($n_{low}$), it emits a photon. The wavelength of this light is characteristic of the element, forming a "line spectrum". For example, the Lyman series involves transitions to $n=1$ and falls in the ultraviolet region, while the Balmer series involves transitions to $n=2$ and falls in the visible region.

Wave-Particle Duality and Photoelectric Effect

Modern physics treats both light and matter as having both wave and particle properties.

• De Broglie Wavelength: Any moving particle has an associated wavelength $\lambda$ given by: $\lambda = \frac{h}{p} = \frac{h}{mv}$ where $h$ is Planck's constant and $p$ is momentum.
• Photoelectric Effect: This phenomenon demonstrates that light behaves as a stream of particles (photons). When light hits a metal, it can eject electrons if the photon energy is greater than the metal's "work function".

Nuclear Chemistry: Decay and Fission

While electronic chemistry deals with the electron cloud, nuclear chemistry focuses on the nucleus, which contains protons and neutrons.

Isotopes, Isotones, and Isoelectronic Species

To navigate nuclear chemistry, one must distinguish between different types of atomic relationships:

• Isotopes: Atoms with the same number of protons (same $Z$) but different numbers of neutrons.
• Isotones: Atoms with the same number of neutrons.
• Isoelectronic Species: Atoms or ions that have the same number of electrons.

Nuclear Reactions and Decay

Nuclei are often unstable and undergo decay to reach a more stable state.

1. Alpha ($\alpha$) Decay: The emission of a helium nucleus ($^4_2\text{He}$), which reduces the atomic number by 2 and the mass number by 4.
2. Beta ($\beta$) Decay: The emission of an electron (or positron), which changes the atomic number but keeps the mass number constant.
3. Nuclear Fission: A heavy nucleus splits into smaller "daughter" nuclei, often accompanied by the release of neutrons. For example, the controlled fission of $^{235}_{92}\text{U}$ can produce $^{142}_{54}\text{Xe}$ and $^{90}_{38}\text{Sr}$ along with several neutrons.

Worked Example: Balancing a Nuclear Equation If $^{238}_{92}\text{U}$ decays into $^{234}_{92}\text{U}$ and an $\alpha$-particle is emitted, what is the other emission?

1. Mass Balance: $238 = 234 + 4$. The mass is balanced.
2. Charge Balance: $92 = 92 + 2 + x$. To balance the charge, $x$ must be $-2$.
3. Conclusion: A charge of $-2$ with no mass change indicates the emission of two $\beta^-$ particles (electrons).

By integrating the study of quantum numbers with the mechanics of nuclear reactions, chemists can predict everything from the color of a flame to the energy produced in a nuclear reactor.