Quantum-Mechanical Model of the Atom Notes

Chapter 7: The Quantum-Mechanical Model of the Atom

Waves

• To understand the electronic structure of atoms, it is essential to understand the nature of electromagnetic radiation.
• Wavelength ($\lambda$): The distance between corresponding points on adjacent waves.

Waves: Frequency and Amplitude

• Frequency ($\nu$): The number of waves passing a given point per unit of time.
• For waves traveling at the same velocity, wavelength and frequency are inversely proportional.
• Amplitude: Directly proportional to intensity.

Electromagnetic Radiation

• All electromagnetic radiation travels at the same velocity, the speed of light (c).
• c = 3.00 \times 10^8 \text{ m/s}
• c = \lambda \nu

The Nature of Energy: Quantization

• The wave nature of light cannot explain the glow of an object when its temperature increases.
• Max Planck's Explanation: Energy comes in packets called quanta.
• E = h\nu

The Nature of Energy: Photoelectric Effect and Planck's Constant

• Einstein's Explanation of the Photoelectric Effect:
  • Energy is proportional to frequency: E = h\nu
  • h is Planck’s constant: h = 6.63 \times 10^{-34} \text{ J-s}

Calculating Energy of a Photon

• If the wavelength of light is known, the energy in one photon (packet) of light can be calculated using the equations:
  • c = \lambda \nu
  • E = h\nu

The Nature of Energy: Continuous vs. Line Spectra

• White light can be split into a continuous spectrum.
• However, observation reveals a line spectrum of discrete wavelengths, not a continuous one.

Rydberg's Equation

• Rydberg determined that for hydrogen, the lines correspond to:
  • \frac{1}{\lambda} = R_H
  • R_H = 1.096776 \times 10^7 \text{ m}^{-1}
  • where n2 > n1
• Rydberg did not know the physical significance of n.

Models of the Atom

• Evolution of atomic models:
  • Hard sphere
  • "Raisin & Plum Pudding"
  • Nuclear
• Question: Where are the electrons located?

Niels Bohr's Model

• Niels Bohr's Postulates: Bohr adopted Planck's assumption to explain atomic phenomena.
  • Electrons in an atom can only occupy certain orbits (corresponding to specific energies).
  • Electrons in permitted orbits possess specific, "allowed" energies, and these energies are not radiated from the atom.
  • Energy is absorbed or emitted when an electron moves from one "allowed" energy state to another; the energy is defined by E = h\nu

Energy Transitions

• The energy absorbed or emitted during electron promotion or demotion can be calculated by:
  • \Delta E = -hcRH\left(\frac{1}{nf^2} - \frac{1}{n_i^2}\right)
  • Where hcR_H = 2.18 \times 10^{-18} \text{ J}
  • ni and nf are the initial and final energy levels of the electron.

Wavelength and Spectral Lines

• Electronic transitions produce spectral lines; the longest wavelength spectral line corresponds to the smallest energy transition.

Energy Emission and Transitions

• Considering only the n = 1 to n = 5 states, the transition that emits the most energy is the one with the largest energy difference between levels.

The Wave Nature of Matter

• Louis de Broglie's Hypothesis: If light can have material properties, then matter should exhibit wave properties.
• Relationship between mass and wavelength:
  • \lambda = \frac{h}{mv}

Wavelength Calculation

• What is the wavelength of a helium-4 ion moving in a cyclotron at 1.0% the speed of light?
  • Mass of helium-4 ion = 6.6 \times 10^{-27} \text{ g}.

The Uncertainty Principle

• Heisenberg's Uncertainty Principle: The more precisely the momentum of a particle is known, the less precisely its position is known.
  • (\Delta x)(\Delta mv) \geq \frac{h}{4\pi}
• In many cases, the uncertainty of an electron's location is greater than the size of the atom itself.

Applicability of the Uncertainty Principle

• The uncertainty principle primarily applies to subatomic particles due to their small size and mass relative to the macroscopic world.

Quantum Mechanics

• Erwin Schrödinger's Contribution: Developed a mathematical treatment incorporating both the wave and particle nature of matter, known as quantum mechanics.

Wave Equation and Probability Density

• The wave equation is designated by the Greek letter psi ($\psi$).
• The square of the wave equation, \psi^2, gives a probability density map indicating the statistical likelihood of finding an electron at any given instant in time but cannot predict an exact location.

Orbitals

• An orbital is a three-dimensional space around a nucleus where an electron is most likely to be found.
• An orbital has a shape representing electron density (not a defined path).
• An orbital can hold up to 2 electrons.

Orbit vs. Orbital

• An orbit is a well-defined circular path, while an orbital is a wave function representing the probability of finding an electron at any point in space.

Quantum Numbers

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

Principal Quantum Number (n)

• The principal quantum number (n) describes the energy level on which the orbital resides.
• Values of n are integers ≥ 1.

Azimuthal Quantum Number (l)

• The azimuthal quantum number (l) defines the shape of the orbital.
• Allowed values of l are integers ranging from 0 to n - 1.
• Letter designations are used to communicate the different values of l and, therefore, the shapes and types of orbitals.
• Also called Orbital Momentum.

Azimuthal Quantum Number Values

• Value of l:
  • 0 corresponds to an s orbital (Sharp).
  • 1 corresponds to a p orbital (Principal).
  • 2 corresponds to a d orbital (Diffuse).
  • 3 corresponds to an f orbital (Fundamental).

Magnetic Quantum Number (ml)

• The magnetic quantum number (m_l) describes the three-dimensional orientation of the orbital.
• Values are integers ranging from -l to l: -l ≤ m_l ≤ l
• On any given energy level, there can be up to 1 s orbital, 3 p orbitals, 5 d orbitals, 7 f orbitals, etc.

Shells and Subshells

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

Energy Level Differences

• The energy difference between n = 1 and n = 2 is greater than between n = 2 and n = 3 because in Bohr's equation, the difference between \frac{-1}{(2)^2} and \frac{-1}{(1)^2} is greater than between \frac{-1}{(3)^2} and \frac{-1}{(2)^2}.

s Orbitals

• Value of l = 0.
• Spherical in shape.
• The radius of the sphere increases with increasing value of n.

s Orbitals and Nodes

• s orbitals possess n-1 nodes, which are regions where there is 0 probability of finding an electron.

p Orbitals

• Value of l = 1.
• Have two lobes with a node between them.
• There are three p orbitals: px, py, and p_z.

p Orbital Features

• The probability of finding an electron in the interior of a p-orbital lobe is greater than it is on the edges.

d Orbitals

• Value of l = 2.
• Four of the five orbitals have 4 lobes; the other resembles a p orbital with a doughnut around the center.

Energies of Orbitals

• For a one-electron hydrogen atom, orbitals on the same energy level have the same energy (degenerate).

Energies of Orbitals in Multi-Electron Atoms

• As the number of electrons increases, so does the repulsion between them.
• Therefore, in many-electron atoms, orbitals on the same energy level are no longer degenerate.

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.

Spin Quantum Number Values

• This led to a fourth quantum number, the spin quantum number, m_s.
• The spin quantum number has only 2 allowed values: +\frac{1}{2} and -\frac{1}{2}.

Pauli Exclusion Principle

• No two electrons in the same atom can have exactly the same energy.
• No two electrons in the same atom can have identical sets of quantum numbers.

Electron Capacity of Orbitals

• A 4d orbital can hold a maximum of 10 electrons because there are five 4d orbitals, and each can contain up to 2 electrons.

Orbitals and Quantum Numbers

• For n = 3 and l = 2, there are 5 orbitals described by these quantum numbers.

Electrons in p Orbitals

• The number of electrons that can occupy a p orbital is 2.
• The number of p orbitals in the 2p sublevel is 3.
• The maximum number of electrons in the n = 3 level is 18.

Electron Configurations

• Definition: Distribution of all electrons in an atom.
• Components:
  • Number denoting the energy level.
  • Letter denoting the type of orbital.
  • Superscript denoting the number of electrons in those orbitals.

Orbital Diagrams

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

Quantum Numbers and Subshells

• The electron subshell 3p represents the principal quantum number n = 3 and azimuthal quantum number l = 1.

Rules for Filling Orbitals

• Aufbau Principle: Fill lowest energy levels first.
• Hund’s Rule: “No seconds until everyone is served.”
• Pauli Exclusion Principle: Every electron is unique.

Hund's Rule Explained

• “For degenerate orbitals, the lowest energy is attained when the number of electrons with the same spin is maximized.”

Electronic Configuration Anomalies

• The element with the electronic configuration [Ar] 4s¹3d5 is Chromium.

Periodic Table and Orbital Filling

• Orbitals are filled in increasing order of energy.
• Different blocks on the periodic table correspond to different types of orbitals.

Order of Filling Orbitals

• The filling order of orbitals: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

Numerical Order for Filling Orbitals

• Fill in numerical order. If there is a tie, fill lowest n first.

Pascal's Tree

• Pascal's Tree is a visual aid for determining the order of filling atomic orbitals: 1s, 2s, 2p 3s, 3p 4s, 3d 4p 5s, 4d 5p 6s, 4f 5d 6p 7s, 5f 6d 7p 8s

Orbital Occupancy

• Based on the structure of the periodic table, the 6s orbital is occupied before the 5d orbitals.

Valence Electrons

• Valence electrons determine the chemical properties of the elements.
• Valence electrons are the electrons in the highest energy level and in any unfilled lower level orbitals.
• Valence electrons are related to the group number of the element.
  • Example: Phosphorus has 5 valence electrons. 1s^2 2s^2 2p^6 3s^2 3p^3

Groups and Valence Electrons

• All the elements in a group have the same number of valence electrons.
  • Example: Elements in group 2A(2) have two (2) valence electrons.

Determining Valence Electrons

• O has 6 valence electrons
• Al has 3 valence electrons
• Cl has 7 valence electrons

Valence Electrons of Vanadium

• The valence electrons of vanadium are 4s^2 3d^3.

Valence Electrons of Gallium

• The valence electrons of gallium are 4s^2 4p^1.

Unpaired Electrons in Selenium

• Selenium has 2 unpaired electrons.

Anomalies in Electron Configurations

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

Chromium Configuration

• The electron configuration for chromium is [Ar] 4s¹ 3d5 rather than the expected [Ar] 4s² 3d4.

Reason for Anomalies

• This occurs because the 4s and 3d orbitals are very close in energy.
• These anomalies occur in f-block atoms, as well.

Unpaired Electrons in Chromium

• Chromium has 6 unpaired electrons.