Electronic Structure of Atoms Notes

Electronic Structure of Atoms

Introduction to Electronic Structure

• Electronic structure refers to the arrangement and energy of electrons within an atom.
• Understanding electronic structure is crucial for comprehending the chemical properties of elements.
• The behavior of extremely small particles, like electrons, can only be explained by understanding wave properties.

Waves and Electromagnetic Radiation

• Electromagnetic radiation travels as waves through space at the speed of light.
• Wavelength ($\lambda$) is the distance between corresponding points on adjacent waves.
• Frequency ($\nu$) is the number of waves passing a given point per unit of time.
• For waves traveling at the same velocity, longer wavelength means smaller frequency.
• The speed of light (c) is constant: $c = 3.00 \times 10^8$ m/s. The relationship between the speed of light, wavelength, and frequency is given by: $c = \lambda\nu$

Electromagnetic Radiation Spectrum

• Different types of electromagnetic radiation have different wavelengths and energies.
• A wide range of units may be used to define wave length.

Limitations of Wave Theory

• Certain observed properties of atoms interacting with electromagnetic radiation cannot be explained by wave theory:
  • Emission of light from hot objects (blackbody radiation).
  • Emission of electrons from metal surfaces when light is shone on them (the photoelectric effect).
  • Emission of light from electronically excited gas atoms (emission spectra).

The Nature of Energy - Quanta

• Max Planck proposed that energy is emitted in discrete packets called quanta.

The Photoelectric Effect

• Einstein explained the photoelectric effect using quanta.
• Each metal has a specific energy threshold for electron ejection.
• Energy is proportional to frequency: $E = h\nu$, where h is Planck’s constant ($h = 6.626 \times 10^{-34}$ J·s).

Atomic Emissions

• Emission spectra observed from atoms and molecules presented a mystery in the early 20th century.

Continuous vs. Line Spectra

• Atoms and molecules produce line spectra, consisting of discrete wavelengths, rather than a continuous spectrum (rainbow).
• Each element has a unique line spectrum.

The Hydrogen Spectrum

• Johann Balmer (1885) discovered a formula relating the four lines in the hydrogen spectrum to integers.
• Johannes Rydberg advanced this formula, introducing the Rydberg constant ($R_H$).
• Neils Bohr explained the mathematical basis for this relationship.

The Bohr Model

• Niels Bohr's model incorporated Planck’s quantum theory.
  1. Electrons in a hydrogen atom are only permitted to occupy orbits with specific radii and corresponding energies.
  2. An electron in a permitted orbit is in an allowed energy state and does not radiate energy or spiral into the nucleus.
  3. Energy is emitted or absorbed only when an electron transitions between energy states. The energy is emitted or absorbed as a photon with energy $E = h\nu$.

Energy States and Transitions

• Ground state: The lowest energy state of an electron.
• Excited state: Any energy state higher than the ground state.
• The energy of each orbit has a specific value related to $R_H$. Transitions between energy levels can be calculated.

Energy Transitions and Photon Emission/Absorption

• Positive $\Delta E$: Energy is absorbed (photon absorbed); occurs when $n<em>f > n</em>i$.
• Negative $\Delta E$: Energy is released (photon emitted); occurs when $n<em>f < n</em>i$.

Limitations of the Bohr Model

• The Bohr model only accurately predicts the spectra of hydrogen.
• It contradicts classical physics, which predicts that an electron should spiral into the nucleus.
• The model assumes circular motion, which is not wave-like.

Important Ideas from the Bohr Model

• Key concepts incorporated into the modern atomic model:
  1. Electrons exist only in discrete energy levels, described by quantum numbers.
  2. Energy is involved when an electron transitions between energy levels.

The Wave Nature of Matter

• Louis de Broglie proposed that if light can have material properties, then matter should exhibit wave properties.
• He related mass and wavelength with the equation: $\lambda = \frac{h}{mv}$, where:
  • $\lambda$ is wavelength
  • $h$ is planck's constant
  • $m$ is mass
  • $v$ is velocity

The Uncertainty Principle

• Heisenberg's Uncertainty Principle: the more precisely the momentum of a particle is known, the less precisely its position is known.

Quantum Mechanics

• Erwin Schrödinger developed a mathematical treatment incorporating both the wave and particle nature of matter, known as quantum mechanics.

Wave Functions and Electron Density

• Solving Schrödinger’s wave equation for hydrogen yields wave functions for the electron.
• The square of the wave function gives the electron density, representing the probability of finding an electron at a given location.

Quantum Numbers

• Solving the wave equation provides a set of wave functions (orbitals) and their corresponding energies.
• Each orbital describes a spatial distribution of electron density.
• An orbital is defined by a set of three quantum numbers.

Principal Quantum Number (n)

• Describes the energy level on which the orbital resides.
• Values are integers greater than or equal to 1 ($n \geq 1$).
• Corresponds to the energy levels in the Bohr model.

Angular Momentum Quantum Number (l)

• Defines the shape of the orbital.
• Allowed values are integers ranging from 0 to $n-1$.
  • $l = 0$ corresponds to an s orbital.
  • $l = 1$ corresponds to a p orbital.
  • $l = 2$ corresponds to a d orbital.
  • $l = 3$ corresponds to an f orbital.

Magnetic Quantum Number ($m_l$)

• Describes the three-dimensional orientation of the orbital.
• Allowed values are integers ranging from $-l$ to $l$, including 0: $-l \leq m_l \leq l$.
• For a given energy level, there can be:
  • Up to 1 s orbital.
  • Up to 3 p orbitals.
  • Up to 5 d orbitals.
  • Up to 7 f orbitals, and so on.

Electron Shells and Subshells

• Orbitals with the same value of n form an electron shell.
• Different orbital types within a shell are called subshells.

s Orbitals

• The value of $l$ for s orbitals is 0.
• They are spherical in shape.
• The radius of the sphere increases with the value of n.

Characteristics of ns Orbitals

• For an ns orbital, the number of peaks is n.
• For an ns orbital, the number of nodes (where there is zero probability of finding an electron) is $n - 1$.
• As n increases, the electron density becomes more spread out, and there is a greater probability of finding an electron farther from the nucleus.

p Orbitals

• The value of $l$ for p orbitals is 1.
• They have two lobes with a node between them.

d Orbitals

• The value of $l$ for a d orbital is 2.
• Four of the five d orbitals have four lobes; the fifth resembles a p orbital with a doughnut around the center.

f Orbitals

• Very complicated shapes.
• Seven equivalent orbitals in a sublevel.
• $l = 3$

Energies of Orbitals—Hydrogen

• For a one-electron hydrogen atom, orbitals on the same energy level have the same energy.
• These are called degenerate orbitals.

Energies of Orbitals—Many-electron Atoms

• As the number of electrons increases, so does the repulsion between them.
• In atoms with more than one electron, not all orbitals on the same energy level are degenerate.
• Orbital sets in the same sublevel are still degenerate.
• Energy levels can overlap (e.g., 4s is lower in energy than 3d).

Spin Quantum Number, ms

• In the 1920s, it was discovered that two electrons in the same orbital do not have exactly the same energy.
• The “spin” of an electron describes its magnetic field, which affects its energy.
• The spin quantum number (ms) has two allowed values: +½ and –½.

Pauli Exclusion Principle

• No two electrons in the same atom can have the same set of four quantum numbers.
• Therefore, no two electrons in the same atom can have the exact same energy.
• Every electron in an atom must differ by at least one of the four quantum number values: n, l, ml, and ms.

Electron Configurations

• Electron configuration describes how electrons are distributed in an atom.
• The most stable organization is the lowest possible energy, called the ground state.
• Each component consists of:
  • A number denoting the energy level.
  • A letter denoting the type of orbital.
  • A superscript denoting the number of electrons in those orbitals (e.g., $4p^5$).

Orbital Diagrams

• Each box in the diagram represents one orbital.
• Half-arrows represent the electrons.
• The direction of the arrow represents the relative spin of the electron.

Hund’s Rule

• “When filling degenerate orbitals, the lowest energy is attained when the number of electrons having the same spin is maximized.”
• For a set of orbitals in the same sublevel, there must be one electron in each orbital before pairing, and the electrons should have the same spin as much as possible.

Condensed Electron Configurations

• Elements in the same group of the periodic table have the same number of electrons in the outermost shell (valence electrons).
• Filled inner shell electrons are called core electrons and include completely filled d or f sublevels.
• A shortened version of an electron configuration is written using brackets around a noble gas symbol, followed by only the valence electrons.

Transition Metals

• Argon (atomic number 18) ends period 3, with an electron configuration of $1s^22s^22p^63s^23p^6$.
• Potassium (atomic number 19) fills 4s before 3d.
• Transition metals in the 4th period involve the filling of 3d orbitals after the 4s orbital is filled.

Lanthanides and Actinides

• Lanthanide elements (atomic numbers 57 to 70) have electrons entering the 4f sublevel.
• Actinide elements (including Uranium, at. no. 92, and Plutonium, at. no. 94) have electrons entering the 5f sublevel.

Periodic Table and Electron Configuration

• Orbitals are filled in increasing order of energy.
• Different blocks on the periodic table correspond to different types of orbitals:
  • s-block: Blue (representative elements)
  • p-block: Pink (representative elements)
  • d-block: Orange (transition elements)
  • f-block: Tan (lanthanides and actinides, or inner transition elements)
• The s and p blocks are called the main-group elements.

Following the Periodic Table

• The periodic table is followed directly when determining the electron configuration for most elements.

Anomalies in Electron Configurations

• Some irregularities occur when there are enough electrons to half-fill s and d orbitals on a given row.

Chromium as an Anomaly

• The electron configuration for chromium is [Ar] $4s^1 3d^5$ rather than the expected [Ar] $4s^2 3d^4$.
• This occurs because the 4s and 3d orbitals are very close in energy.
• Similar anomalies occur in f-block atoms with f and d orbitals.