In-Depth Notes on Electronic Structure of Atoms
Electronic Structure of Atoms
Introduction to Electronic Structure
Nucleus Composition: An atom's nucleus consists of protons and neutrons, while electrons orbit around this nucleus.
Size Comparison: Electrons are approximately 1800 times smaller in mass than protons.
Wave Properties: Small particles can be described using wave properties in physics.
Understanding Electromagnetic Radiation
Nature of Electromagnetic Radiation: To comprehend an atom's electronic structure, one must understand electromagnetic radiation, which propagates as waves at the speed of light.
Key Terms
Wavelength ($\lambda$): Distance between corresponding points on adjacent waves.
Frequency ($v$): Number of waves that pass a given point in a specific time frame.
Relationship: For waves traveling at the same velocity, longer wavelengths correlate to smaller frequencies.
Properties of Electromagnetic Radiation
Speed of Light: All electromagnetic radiation travels at the speed of light, $c = 3 \times 10^8 \, m/s$.
Wavelength-Frequency Relationship: The relationship is given by the equation ( c = \lambda v ).
Visible Spectrum: Humans can only perceive a narrow region of the electromagnetic spectrum.
Types of Electromagnetic Radiation
Common Wavelength Units:
Angstrom ($10^{-10}$ m): X-ray
Nanometer (nm, $10^{-9}$ m): Ultraviolet, visible
Micrometer ($\mu m$, $10^{-6}$ m): Infrared
Millimeter (mm, $10^{-3}$ m): Microwave
Centimeter (cm, $10^{-2}$ m): Microwave
Meter (m): Television, radio
Kilometer (km, $10^{3}$ m): Radio
Diversity in Radiation: Types of electromagnetic radiation vary in wavelength and energy.
Quantized Energy and Photons
Key Observations of Atomic Interaction with Electromagnetic Radiation:
Black Body Radiation: The light emitted from heated objects.
Photoelectric Effect: Emission of electrons when light strikes a metal surface.
Emission Spectra: Light emitted from excited gas atoms.
Wave Limitations: These phenomena cannot be solely explained through wave theories.
Black Body Radiation
Emission Mechanism: An object emits light when its temperature increases, which classical physics fails to explain adequately, predicting emissions of UV and X-rays that are not observed.
Concept of Quanta
Max Planck: Proposed energy exists in discrete packets called quanta (singular: quantum).
Analogy: Quanta can be likened to stairs (discrete) versus a ramp (continuous).
Photoelectric Effect
Einstein's Explanation: Quanta explain the photoelectric effect, where specific metals emit electrons at certain energy levels. The relation is expressed asE = h v
Where (h) is Planck's constant (6.626 \times 10^{-34} \, J \, s).
Atomic Emission of Gas
Emission Spectra Observations: Different line spectra are emitted from energy observed from gas atoms.
Line Spectra and the Bohr Model
Discrete Spectra: Only discrete wavelengths are observed for elements instead of continuous spectra.
Unique Spectra: Each element has a unique line spectrum.
The Hydrogen Spectrum and Rydberg's Formula
Mathematical Relationships: Rydberg derived formulas correlating wavelength with integers, termed Rydberg's constant ($R_H$).
Bohr's Assumptions: Energy exists in small packets (quanta), as described by Planck.
The Bohr Model (Hydrogen Atom)
Allowed Orbits: Only certain orbits (specific radii) are permitted.
Energy States: An electron in an allowed orbit is in an allowed energy state and does not radiate energy.
Energy Transitions: Energy is emitted or absorbed as the electron changes states, expressed as:
E = h v
Ground State and Excited State
Lowest Energy State: Electrons in the lowest energy state are termed the ground state (n=1).
Higher Energy States: Energy states are termed excited state (n>1).
Energy Transition Calculation
Energy Change: Calculating energy change between levels:
\Delta E = Ef - Ei = -2.18 \times 10^{-18} J \left( \frac{1}{nf^2} - \frac{1}{ni^2} \right)
Energy and Photon Emission
Photon Emission: A positive (\Delta E) indicates energy absorption (photon absorbed) while a negative (\Delta E) indicates energy release (photon emitted).
Limitations of the Bohr Model
Hydrogen Specific: Functional only for hydrogen (one electron).
Assumption: Electrons do not spiral into the nucleus, a reality contradicting classical physics.
Wave Properties: Electrons exhibit wave-like properties that the model does not address.
Important Ideas from the Bohr Model
Discrete Energy Levels: Electrons occupy specific energy levels defined by quantum numbers.
Energy Transitions: Energy involvement is crucial in electron state transitions.
Wave Behavior of Matter
Louis de Broglie: Proposed that if light can exhibit material properties, matter should demonstrate wave properties, with the relationship given by:
\lambda = \frac{h}{mv}
Uncertainty Principle
Heisenberg's Conclusion: The more accurately the momentum is known, the less accurately the position can be determined:
\Delta x \Delta p \geq \frac{h}{4\pi}
Quantum Mechanics and Atomic Orbitals
Schrödinger's Wave Equation: Provides a mathematical description of wave-particle duality, yielding wave functions for electrons. The square of a wave function gives electron density probabilities.
Orbitals and Quantum Numbers
Wave Functions: Solving the wave equation defines orbitals and their energies; an orbital represents the spatial distribution of electron density.
Electron Capacity: Each orbital can hold a maximum of two electrons.
Principal Quantum Number (n)
Energy Level Description: Positive integral values corresponding to energy levels; as n increases, orbitals become larger and the electrons are less tightly bound.
Angular Momentum Quantum Number (l)
Orbital Shape: Defines orbital shapes with integer values ranging 0 to n-1. Symbols for l:
s (l=0): Spherical
p (l=1): Dumbbell-shaped
d (l=2): Cloverleaf-shaped
f (l=3): Complex shapes
Magnetic Quantum Number (mₗ)
Orbital Orientation: Describes the 3D orientation of orbitals. Varies from -l to l, allowing multiple orbital configurations.
Quantum Numbers Summary
Electron Shells: Orbitals with the same n value form an electron shell. Subshells refer to different orbital types within the same energy level.
Representation of Orbitals
s Orbitals: Spherical shape; number of peaks corresponds to n; number of nodes (zero probability of finding electrons) is n-1.
p Orbitals: Two lobes with one node between them.
d Orbitals: Complex shapes with four of five d orbitals displaying lobes and one resembling a p orbital with a doughnut.
f Orbitals: Very complex with seven equivalent orbitals.
Hydrogen Atom Orbital Energies
All orbitals at the same energy level are degenerate for one-electron hydrogen atoms.
Many-Electron Atoms
In multi-electron atoms, electron repulsion causes orbital energy levels to vary; not all orbitals at the same energy level are degenerate.
Spin Quantum Number (mₛ)
Electron Spin: Two electrons in the same orbital exhibit different energies due to their spins, which can only take values of +1/2 or -1/2.
Pauli Exclusion Principle
Quantum Number Uniqueness: No two electrons can have identical sets of four quantum numbers, necessitating differences across at least one quantum number.
Electron Configurations
Definition: Distribution of electrons within an atom; the most stable arrangement occurs at the lowest energy state.
Components: Each electron configuration includes a number (energy level), a letter (type of orbital), and superscript (number of electrons).
Orbital Diagrams
Visualization: Orbital diagrams depict orbitals and their electron fills, utilizing arrows denoting electron spin direction, adhering to Hund's Rule and Pauli's Exclusion Principle.
Condensed Electron Configurations
Core and Valence Electrons: Core electrons are primarily from noble gases; only valence electrons are listed in condensed electronic configurations.
Transition Metals and Anomalies
Electron Filling Sequence: Transition metals fill the 4s orbital before the 3d, with special configurations observed during half-filled or filled states.
Examples: Chromium exhibits anomalies in electron configurations due to energy proximity of the 4s and 3d orbitals.
Conclusion
Understanding the electronic structure of atoms requires the combination of concepts from classical and quantum physics, resulting in a more comprehensive model of atomic behavior.