3.1 Hydrogen atom and quantum numbers

The quantum mechanical model of the hydrogen atom is a game-changer in understanding atomic structure. It uses the Schrödinger equation to describe electron behavior as probability distributions called atomic orbitals. This model explains discrete energy levels and spectral lines.

Quantum numbers are the secret sauce of atomic structure. They describe an electron's energy, shape, orientation, and spin. The four quantum numbers - principal, angular momentum, magnetic, and spin - uniquely identify each electron and determine its properties in an atom.

Quantum Mechanical Model of Hydrogen

Schrödinger Equation and Atomic Orbitals

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• The quantum mechanical model of the hydrogen atom is based on the Schrödinger equation, which describes the electron's behavior in terms of probability distributions called atomic orbitals
• Atomic orbitals represent the regions in space where the electron is most likely to be found
• The electron in a hydrogen atom can only exist in specific, quantized energy states determined by the principal quantum number ($n$), leading to a discrete energy level structure
  • The allowed energy levels are given by the equation: $E_n = -\frac{13.6 eV}{n^2}$, where $n$ is a positive integer (1, 2, 3, ...)

Implications and Applications

• The quantum mechanical model explains the stability of the hydrogen atom and the existence of spectral lines, which result from transitions between different energy levels
  • When an electron transitions from a higher energy level to a lower one, a photon with a specific wavelength is emitted, corresponding to a spectral line
• The Heisenberg uncertainty principle states that the position and momentum of an electron cannot be simultaneously determined with absolute precision, leading to a probabilistic description of the electron's location
  • This means that the electron's exact path cannot be traced, and its motion is described by a wave function
• The quantum mechanical model of the hydrogen atom serves as a foundation for understanding the structure and properties of more complex atoms and molecules
  • It provides a framework for explaining the periodic table, chemical bonding, and the behavior of electrons in solids

Quantum Numbers and Atomic Structure

Four Quantum Numbers

• The principal quantum number ($n$) determines the main energy level and the average distance of the electron from the nucleus. It can take positive integer values (1, 2, 3, ...)
  • Larger $n$ values correspond to higher energy levels and larger orbital sizes
• The angular momentum quantum number ($l$) describes the shape of the orbital and the magnitude of the electron's angular momentum. It can take integer values from 0 to ($n-1$) and is denoted by letters (s, p, d, f, ...)
  • $l=0$ corresponds to s orbitals, $l=1$ to p orbitals, $l=2$ to d orbitals, and $l=3$ to f orbitals
• The magnetic quantum number ($m_l$) specifies the orientation of the orbital in space relative to an external magnetic field. It can take integer values from $-l$ to $+l$, including 0
  • For example, a p orbital ($l=1$) has three possible orientations: $m_l=-1, 0, +1$
• The spin quantum number ($m_s$) describes the intrinsic angular momentum (spin) of the electron. It can have two values: $+\frac{1}{2}$ (spin up) and $-\frac{1}{2}$ (spin down)
  • The spin quantum number is essential for understanding the Pauli exclusion principle and the electronic configuration of atoms

Uniquely Identifying Electrons

• The combination of these four quantum numbers uniquely identifies each electron in an atom and determines its energy, shape, orientation, and spin
  • No two electrons in an atom can have the same set of four quantum numbers, according to the Pauli exclusion principle
• The quantum numbers provide a complete description of the electron's state in the hydrogen atom and form the basis for understanding the electronic structure of more complex atoms

Energy Levels and Orbitals in Hydrogen

Energy Levels and Subshells

• The energy levels of the hydrogen atom are determined by the principal quantum number ($n$). As $n$ increases, the energy of the electron increases, and the electron occupies higher energy levels (shells)
  • The ground state of hydrogen has $n=1$, while excited states have $n=2, 3, 4, ...$
• Each energy level (shell) contains a specific number of subshells, which are determined by the angular momentum quantum number ($l$). The number of subshells in a given shell is equal to the principal quantum number ($n$)
  • For example, the first shell ($n=1$) has only one subshell (1s), while the second shell ($n=2$) has two subshells (2s and 2p)
• The subshells are denoted by letters: s ($l=0$), p ($l=1$), d ($l=2$), f ($l=3$), and so on
  • The letter designations originate from spectroscopic notation: sharp, principal, diffuse, and fundamental

Orbitals and Electron Spin

• The magnetic quantum number ($m_l$) determines the number of orbitals within each subshell. There are ($2l+1$) orbitals in each subshell, each with a specific orientation in space
  • For instance, the p subshell ($l=1$) has three orbitals: p_x ($m_l=-1$), p_y ($m_l=0$), and p_z ($m_l=+1$)
• The spin quantum number ($m_s$) indicates the spin state of the electron in each orbital, which can be either spin up ($+\frac{1}{2}$) or spin down ($-\frac{1}{2}$), following the Pauli exclusion principle
  • Each orbital can accommodate a maximum of two electrons with opposite spins
• The energy levels and orbitals of the hydrogen atom form the basis for understanding the electronic structure of more complex atoms and the periodic table

Shapes and Orientations of Atomic Orbitals

s, p, and d Orbitals

• The shape and orientation of atomic orbitals are determined by the angular momentum quantum number ($l$) and the magnetic quantum number ($m_l$)
• The s orbitals ($l=0$) are spherically symmetric and have no angular dependence. They have a single orientation ($m_l=0$)
  • The probability density of an electron in an s orbital is highest at the nucleus and decreases radially outward
• The p orbitals ($l=1$) have three possible orientations ($m_l=-1, 0, +1$) and are shaped like dumbbells. They are designated as p_x, p_y, and p_z, indicating their alignment along the x, y, and z axes, respectively
  • The p orbitals have two lobes with opposite signs, separated by a nodal plane
• The d orbitals ($l=2$) have five possible orientations ($m_l=-2, -1, 0, +1, +2$) and more complex shapes. They are often described as cloverleaf (d_xy, d_xz, d_yz) or dumbbell with a torus (d_x^2-y^2, d_z^2) shapes
  • The d orbitals have more intricate nodal surfaces and are important in describing the electronic structure of transition metals

f Orbitals and Implications for Bonding

• The f orbitals ($l=3$) have seven possible orientations ($m_l=-3, -2, -1, 0, +1, +2, +3$) and even more intricate shapes, which are challenging to visualize and describe
  • The f orbitals are relevant in the electronic structure of lanthanides and actinides
• The shapes and orientations of atomic orbitals have significant implications for chemical bonding and the formation of molecules, as they determine the spatial distribution of electrons and the directionality of bonds
  • The overlap of atomic orbitals leads to the formation of molecular orbitals, which govern the chemical and physical properties of molecules
• Understanding the shapes and orientations of atomic orbitals is crucial for predicting the geometry of molecules, the strength of chemical bonds, and the reactivity of atoms and molecules