Quantum Mechanics: De Broglie and Uncertainty

De Broglie Wavelength

• Every particle has a wave nature and can be described by the De Broglie Wavelength (λ).
• The De Broglie wave equation is λ = h/mv (where h is Planck's constant, m is mass, and v is velocity).

Heisenberg Uncertainty Principle

• The wave and particle nature of an electron are complementary, meaning if you know more about one, you know less about the other.
• Heisenberg's Uncertainty Principle states that (Δx)(mΔv) ≥ h/4π, where Δx is the uncertainty in position, m is mass, Δv is the uncertainty in velocity, and h is Planck's constant.
• The position of an object is a particle property, and the velocity of an object is related to its wave nature.
• For an electron in an atom, the uncertainty in position (Δx) is roughly the size of the atom.
• Unlike the Bohr atom model with fixed electron orbits, the quantum-mechanical atom only allows us to determine the probability of finding an electron in a specific region around the nucleus due to the Heisenberg Uncertainty Principle.
• Future electron positions can only be described statistically and cannot be precisely calculated.

The Schrödinger Equation

• The Schrödinger equation (ꞪΨ = EΨ) can be used to solve for electron positions within an atom by generating a wavefunction (Ψ).
• Ɦ is the Hamiltonian operator, which involves solving a differential equation.
• E is the total energy of the electron.
• The wavefunction (Ψ) is called an orbital.
• The square of the wavefunction (Ψ2) represents the probability of finding an electron at a given location.

Quantum Numbers

• The principal quantum number (n) represents the energy level and the most probable distance from the nucleus. Its values are 1, 2, 3, etc.
• The energy of an electron in the hydrogen atom only depends on n.
• Energy Levels get closer together as n increases.
• The angular momentum quantum number (l) describes the shape of the orbital and takes on values of 0, 1, 2, ... (n - 1).
• It is also represented by letters: l = 0 (s orbital), l = 1 (p orbital), l = 2 (d orbital), l = 3 (f orbital).
• The magnetic quantum number (ml) describes the spatial orientation of the orbital and takes on values from -l to +l, including 0.
• The number of orbitals in a given energy level (n) is equal to n2.

Shells and Sub-Shells

• Shells are groups of orbitals with the same principal quantum number (n).
• Sub-shells are sets of orbitals within a shell with the same angular momentum quantum number (l).

Shapes of Atomic Orbitals

• s orbitals (l = 0) are spherical.
• The 1s orbital is the most closely related to the nucleus.
• As the principal quantum number (n) increases, the number of radial nodes increases. Radial nodes are points where the probability of finding an electron is zero.
• p orbitals (l = 1) have two lobes of electron density on either side of the nucleus separated by an angular node.
• There are three p orbitals (ml = -1, 0, +1), corresponding to different spatial orientations.
• d orbitals (l = 2) have more complex shapes and there are five d orbitals (ml = -2, -1, 0, +1, +2).
• Radial probability describes the total probability of finding an electron at any point along a given radius.

Electron Spin and the Pauli Exclusion Principle

• The electron spin is an intrinsic property of an electron and can be considered as a magnetic dipole moment.
• The Stern-Gerlach experiment demonstrated electron spin.
• The Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers; this means each orbital can hold a maximum of two electrons with opposite spins.

Electron Configurations

• Aufbau principle: Electrons are filled into orbitals in order of increasing energy: 1s, 2s, 2p, 3s, 3p, etc.
• Hund's Rule: When orbitals of equal energy are available (like the three 2p orbitals), electrons first occupy these orbitals singly before pairing up in the same orbital.
• Condensed electron configurations use the noble gas configuration preceding the element to simplify the notation.
• Exceptions to the Aufbau principle occur due to the close energy difference between 3d ad 4s orbitals in certain transition metals.

Electron Configurations of Ions

• Cations are formed by removing electrons from the highest energy level (n) and the highest value of angular momentum quantum number (l) for that level.
• Anions are formed by adding electrons to the highest energy level (n).

Magnetic Properties

• Paramagnetic materials are attracted to a magnetic field; they contain unpaired electrons.
• Diamagnetic materials are not attracted to a magnetic field; they lack unpaired electrons.

Penetration, Shielding, and Sublevel Energy Splitting

• Penetration is the extent to which an electron in a higher-energy orbital can occupy the space of electrons lower-energy orbitals.
• The greater the penetration, the less shielded the outer electron is, and the higher the effective nuclear charge.
• Penetration and shielding affect the order of energy levels, and the energy levels within a principal shell are not always equal in multi-electron atoms.
• Penetration ability follows the trend: s > p > d.

Summary

• Quantum mechanics provides a framework for understanding the behavior of electrons within atoms.
• The Schrödinger equation and quantum numbers allow us to describe the properties and locations of atomic orbitals.
• Electron spin and Hund's rule play important roles in shaping electron configurations.
• The magnetic behavior of atoms or ions can be explained by the presence or absence of unpaired electrons.