Electronic Structure of Atoms

Electronic Structure of Atoms

Electromagnetic Radiation (EMR)

• To understand the electronic structure of atoms, it's essential to understand electromagnetic radiation (EMR), which is a form of energy transmission through space.

• Electromagnetic waves possess characteristic wavelengths (\lambda) and frequencies (v), which determine their energy and behavior.

• Wavelength (\lambda): The distance between corresponding points on adjacent waves (e.g., crest to crest or trough to trough), typically measured in meters (m) or nanometers (nm).

• Frequency (v): The number of cycles (oscillations) that pass a fixed point per unit of time, usually measured in Hertz (Hz), where 1 Hz = 1 cycle/second.

Waves

• Wavelength: Represented by the symbol \lambda (lambda).

• Amplitude: The height of the wave, which is related to the intensity or brightness of the radiation.

• Relationship between wavelength and frequency:

  • Long wavelength (\lambda) implies low frequency (
    u). This means that waves with longer distances between crests have fewer cycles passing a point per second.

  • Short wavelength (\lambda) implies high frequency (
    u). Conversely, waves with shorter distances between crests have more cycles passing a point per second.

Wave Nature of Light

• All electromagnetic radiation (EMR) types travel through a vacuum at the speed of light (c), which is a fundamental constant.

• c = 3.00 \times 10^8 \text{ m/s}

• Relationship between speed of light, wavelength, and frequency: c = \lambda \nu. This equation shows that the speed of light is the product of the wavelength and frequency of the electromagnetic radiation.

• SI unit of \lambda is the meter (m).

• SI unit of \nu is s-1 (1 s-1 = 1 Hertz = 1 Hz).

Wavelength-Frequency Relationship

• Wavelength (\lambda) and frequency (
  u) have an inverse relationship: \nu \propto 1/\lambda. As one increases, the other decreases proportionally.

Electromagnetic Radiation (EMR) Equations

• All EMR travels at the speed of light (c).

• c = \lambda \nu

• Energy equation: E = h\nu = \frac{hc}{\lambda}, where h is Planck’s constant (6.626 \times 10^{-34} \text{ J.s}). This equation relates the energy of a photon to its frequency and wavelength.

• Relationships:

  • \nu \propto 1/\lambda

  • E \propto \nu

  • E \propto 1/\lambda

• The longer the wavelength (\lambda), the lower the frequency (\nu) and the lower the energy (E).

Electromagnetic Spectrum

• The electromagnetic spectrum encompasses a wide range of electromagnetic radiation types, from low-energy radio waves to high-energy gamma rays.

• High-energy radiations, like X-rays, have much shorter wavelengths than low-energy radiations, like radio waves.

• X-rays possess high energy, sufficient to cause tissue damage and even cancer. Thus, precautions are necessary when using X-rays in medical or industrial applications.

Common Wavelength Units

Quantization of Energy

• A quantum is the smallest amount of energy (photon) that can be emitted or absorbed as electromagnetic radiation. Energy is not continuous but comes in discrete packets.

• Relationship between energy and frequency: E = h\nu, where h is Planck’s constant (6.626 \times 10^{-34} \text{ J.s}). This relationship is fundamental in quantum mechanics.

Photoelectric Effect and Photons

• The photoelectric effect is the emission of electrons from metal surfaces when light shines on them. This phenomenon cannot be explained by classical physics.

• It provides evidence for the particle nature of light and for quantization. Classical physics treated light as a wave, but the photoelectric effect showed that light can also behave as particles (photons).

• Einstein proposed that light travels in energy packets called photons.

• The energy of one photon is E = h\nu.

• Electrons are ejected only if the photons have sufficient energy (i.e., the frequency of the light is above a certain threshold). If the photon energy is less than the binding energy of the electron, no electron is emitted, regardless of the intensity of the light.

Quantum Theory-Quantization of Energy

• Quantized: An electron requires a specific amount of energy (equal to the energy difference between two levels) to transition from one energy level to another. The energy levels are discrete and not continuous.

• This energy can come from heating (thermal energy) or light (EMR).

• Quantum: The smallest amount of energy that can be emitted or absorbed as EMR. It is the fundamental unit of energy in quantum mechanics.

• Photon: A quantum of electromagnetic radiation. It is a particle of light with zero mass and specific energy, frequency, and wavelength.

• Energy of a photon: E = h\nu or E = \frac{hc}{\lambda}. h = 6.626 \times 10^{-34} \text{ J.s} (Planck’s constant)

Quantized States

• Quantized states: Discrete energy levels (e.g., steps). Electrons can only exist at specific energy levels, and transitions between these levels involve the absorption or emission of energy in discrete amounts.

• Unquantized states: Smooth transition between levels (e.g., ramp). In classical physics, energy levels are continuous, allowing for smooth transitions.

Bohr’s Model

• Rutherford's model suggested electrons orbit the nucleus like planets orbit the sun. However, this model had significant problems according to classical physics.

• However, according to classical physics, a charged particle (electron) moving in a circular path should continuously lose energy and spiral into the nucleus. This clearly doesn't happen, as atoms are stable.

• Bohr Model addressed the problems with Rutherford's model by incorporating the concept of quantized energy levels.

• Postulates of Bohr's Model:

  1. Electrons in an atom can only occupy certain discrete orbits (energy levels) around the nucleus. These orbits are associated with specific, quantized energy levels.

  2. Electrons can only move from one allowed orbit to another by absorbing or emitting energy in the form of photons. The energy of the photon must be equal to the energy difference between the two orbits.

  3. The energy of the electron in a particular orbit is given by: E = -2.178 \times 10^{-18}
     ewline J (Z^2/n^2), where Z is the atomic number and n is the principal quantum number (n = 1, 2, 3, …). This equation shows that the energy of an electron is quantized and depends on the principal quantum number.

Energy Transitions

• When an electron transitions from a higher energy level (ni) to a lower energy level ( nf), energy is released in the form of a photon.

• The energy of the emitted photon is equal to the difference in energy between the two levels: \Delta E = Ef - Ei. Because E = h\nu = \frac{hc}{\lambda}, we can find the frequency and wavelength of the photon.

• Change in energy: \Delta E = -2.178
  \times 10^{-18} J (\frac{1}{nf^2} - \frac{1}{ni^2})

  • When an electron is Ionized (removed from n = 1 to n = \infty):

    • \Delta E = E\infty - E1 = 0 - (-2.178 \times 10^{-18} J (Z^2/1^2)) = 2.178 \times 10^{-18} J (Z^2)

Limitations of Bohr’s Model

• Bohr's model was a significant step in understanding atomic structure, but it had limitations:

  • It only accurately predicted the spectra of hydrogen and other single-electron species (like He+ and Li2+).

  • It failed to predict the spectra of multi-electron atoms due to the increased complexity of electron-electron interactions.

  • It violated the Heisenberg Uncertainty Principle, which states that it is impossible to know both the exact position and momentum of an electron simultaneously.

The Wave Mechanical Model

• Louis de Broglie proposed that if light can behave as a particle, then matter (like electrons) can also exhibit wave-like properties.

• de Broglie’s equation relates the wavelength of a particle to its momentum: \lambda = \frac{h}{mv}, where m is the mass of the particle and v is its velocity. This equation implies that particles, like electrons, have a characteristic wavelength that depends on their momentum.

The Uncertainty Principle

• Heisenberg’s Uncertainty Principle: It is fundamentally impossible to know both the position and momentum of a particle with high accuracy simultaneously.

• (Δx)(Δp) ≥ \frac{h}{4π}, where (Δx) is the uncertainty in position and (Δp) is the uncertainty in momentum. This principle implies that the more accurately we know the position of an electron, the less accurately we know its momentum, and vice versa.

Electron Density

• Since we cannot know the exact path of an electron, we use probability distributions to describe the likelihood of finding an electron in a particular region of space.

• Electron density represents the probability of finding an electron at a specific location around the nucleus. Regions with high electron density indicate a high probability of finding the electron, while regions with low electron density indicate a low probability.

Schrödinger’s Equation

• Schrödinger’s equation is a fundamental equation in quantum mechanics that describes the behavior of electrons in atoms and molecules.

• \hat{H}\Psi = E\Psi, where \hat{H} is the Hamiltonian operator, \Psi is the wave function, and E is the energy of the electron. Solving Schrödinger’s equation gives the wave functions (orbitals) and corresponding energy levels for the electron.

Quantum Numbers

• Quantum numbers are a set of numbers that describe the properties of atomic orbitals and the electrons within them. There are four main quantum numbers:

  1. Principal Quantum Number (n):

  • Describes the energy level of the electron and the size of the orbital.

  • Values: n = 1, 2, 3, … (positive integers).

  • Larger n values indicate higher energy levels and larger orbitals.

  1. Angular Momentum Quantum Number (l):

  • Describes the shape of the orbital and the angular momentum of the electron.

  • Values: l = 0, 1, 2, …, (n-1).

  • l = 0 corresponds to an s orbital (spherical shape).

  • l = 1 corresponds to a p orbital (dumbbell shape).

  • l = 2 corresponds to a d orbital (more complex shape).

  • l = 3 corresponds to an f orbital (even more complex shape).

  1. Magnetic Quantum Number (m_l):

  • Describes the orientation of the orbital in space.

  • Values: m_l = -l, -l+1, …, 0, …, l-1, l (integers from -l to +l).

  • For a given l, there are 2l + 1 possible values of m_l, corresponding to the number of orbitals with that shape.

  • For example, if l = 1 (p orbital), m_l can be -1, 0, or +1, indicating three p orbitals oriented along the x, y, and z axes.

  1. Spin Quantum Number (m_s):

  • Describes the intrinsic angular momentum of the electron, which is quantized and called spin.

  • Values: m_s = +1/2 or -1/2, often referred to as spin-up and spin-down.

  • This quantum number accounts for the two possible spin states of an electron, which contribute to the magnetic properties of atoms.

Shapes of Atomic Orbitals

• Orbitals are mathematical functions that describe the behavior of electrons in atoms. The shapes of atomic orbitals are determined by the angular momentum quantum number (l).

  • s orbitals (l = 0) are spherical and have no angular dependence. The electron density is highest at the nucleus and decreases with increasing distance from the nucleus.

  • p orbitals (l = 1) have a dumbbell shape and are oriented along the x, y, and z axes. They have one node at the nucleus, where the electron density is zero.

  • d orbitals (l = 2) have more complex shapes with two nodal planes. There are five d orbitals, each with a different spatial orientation.

  • f orbitals (l = 3) have even more complex shapes with three nodal surfaces. There are seven f orbitals, each with a different spatial orientation.

Energy Level Diagram

• The energy level diagram shows the relative energies of the atomic orbitals in an atom. In general, the energy of an orbital increases with increasing principal quantum number (n).

• Within a given energy level (n), the energy of the orbitals increases with increasing angular momentum quantum number (l): s < p < d < f.

• Electrons fill the orbitals in order of increasing energy, following the Aufbau principle, Hund’s rule, and the Pauli exclusion principle.

Electron Configuration

• Electron configuration describes how electrons are distributed among the various atomic orbitals in an atom. It provides detailed information about the electronic structure of an atom and is essential for understanding its chemical properties.

• The electron configuration is written using the principal quantum number (n), the angular momentum quantum number (l), and the number of electrons in each orbital.

• For example, the electron configuration of hydrogen (H) is 1s1, indicating that it has one electron in the 1s orbital.

Rules for Writing Electron Configurations

• Aufbau Principle: Electrons fill the orbitals in order of increasing energy. Lower energy orbitals are filled before higher energy orbitals.

• Hund’s Rule: Within a given subshell (orbitals with the same n and l), electrons are distributed among the orbitals in such a way as to maximize the number of unpaired electrons. This means that electrons will individually occupy each orbital within a subshell before doubling up in any one orbital.

• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers (n, l, ml, ms). This means that each orbital can hold a maximum of two electrons, and they must have opposite spins (+1/2 and -1/2).

Orbital Diagrams and Hund’s Rule

• Orbital diagrams are visual representations of the electron configuration of an atom. They show the orbitals as boxes or circles and the electrons as arrows.

• Hund’s rule states that electrons are distributed among the orbitals within a subshell in such a way as to maximize the number of unpaired electrons. This means that electrons will individually occupy each orbital within a subshell before doubling up in any one orbital, and all unpaired electrons will have the same spin.

Condensed Electron Configurations

• Condensed electron configurations provide a shorthand notation for writing electron configurations of atoms. They use the symbol of the noble gas that precedes the element in the periodic table to represent the core electrons, followed by the valence electrons.

• For example, the condensed electron configuration of sodium (Na) is [Ne] 3s1, indicating that it has the same core electron configuration as neon (Ne) plus one additional electron in the 3s orbital.

Valence Electrons

• Valence electrons are the electrons in the outermost shell (highest principal quantum number n) of an atom. These electrons are primarily responsible for the chemical properties of the atom because they are involved in chemical bonding.

• The number of valence electrons determines the group number of an element in the periodic table.

Ion Formation

• Ion formation involves the gain or loss of electrons from an atom, resulting in the formation of charged species called ions.

• When an atom loses electrons, it forms a positive ion called a cation. Metals typically lose electrons to form cations.

• When an atom gains electrons, it forms a negative ion called an anion. Nonmetals typically gain electrons to form anions.

Ion Electron Configurations

• The electron configurations of ions are determined by adding or removing electrons from the valence orbitals of the neutral atom.

• When forming cations, electrons are removed from the valence orbitals in order of increasing energy. For example, when iron (Fe) forms Fe2+, it loses two electrons from its 4s orbital.

• When forming anions, electrons are added to the valence orbitals according to the Aufbau principle and Hund’s rule. For example, when chlorine