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- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/06%3A_The_Hydrogen_Atom/6.04%3A_Hydrogen_Atomic_Orbitals_Depend_upon_Three_Quantum_NumbersThis page covers wavefunctions of the hydrogen atom, highlighting the role of quantum numbers n, l, and ml in determining electron position and probability density. It discusses normaliza...This page covers wavefunctions of the hydrogen atom, highlighting the role of quantum numbers n, l, and ml in determining electron position and probability density. It discusses normalization, orthogonality, allowed values for the quantum numbers, and the radial behavior of wavefunctions related to atomic number Z.
- https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/05%3A_Electrons_in_Atoms/5.13%3A_OrbitalsThis page discusses electron configurations and orbitals defined by quantum numbers, detailing the four types: s (spherical), p (dumbbell-shaped), d (complex), and f (most complex, wit...This page discusses electron configurations and orbitals defined by quantum numbers, detailing the four types: s (spherical), p (dumbbell-shaped), d (complex), and f (most complex, with seven shapes). It also compares the regulations preventing plane collisions to the restrictions on electron positioning.
- https://chem.libretexts.org/Courses/Western_Washington_University/Biophysical_Chemistry_(Smirnov_and_McCarty)/01%3A_Biochemical_Thermodynamics/1.05%3A_The_Boltzmann_Distribution_and_the_Statistical_Definition_of_EntropyIn this chapter we introduce the statistical definition of entropy as formulated by Boltzmann. This allows us to consider entropy from the perspective of the probabilities of different configurations ...In this chapter we introduce the statistical definition of entropy as formulated by Boltzmann. This allows us to consider entropy from the perspective of the probabilities of different configurations of the constituent interacting particles in an ensemble. This conception of entropy led to the development of modern statistical thermodynamics. For systems that can exchange thermal energy with the surroundings, the equilibrium probability distribution will be the Boltzmann distribution.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/09%3A_Chemical_Bonding_in_Diatomic_Molecules/9.05%3A_Bonding_and_Antibonding_OrbitalsThis page explores the molecular orbitals of the H+ ion using the LCAO method, which includes bonding and antibonding characteristics. The bonding σ1s orbital stabilizes the sy...This page explores the molecular orbitals of the H+ ion using the LCAO method, which includes bonding and antibonding characteristics. The bonding σ1s orbital stabilizes the system through constructive interference, while the antibonding σ∗1s destabilizes it through destructive interference.
- https://chem.libretexts.org/Courses/University_of_North_Carolina_Charlotte/CHEM_2141%3A__Survey_of_Physical_Chemistry/10%3A_Molecular_Spectroscopy/10.05%3A_Vibrational_OvertonesThis page discusses the limitations of the harmonic oscillator model for molecular vibrations, particularly at higher energy levels where anharmonic effects become significant. The anharmonic oscillat...This page discusses the limitations of the harmonic oscillator model for molecular vibrations, particularly at higher energy levels where anharmonic effects become significant. The anharmonic oscillator model, incorporating higher-order terms, offers more accurate predictions and allows for transitions between various vibrational states, resulting in overtones. Observed frequencies align better with anharmonic models, especially for higher energy levels, leading to weaker intensity lines.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/01%3A_The_Dawn_of_the_Quantum_Theory/1.08%3A_The_Bohr_Theory_of_the_Hydrogen_AtomThis page examines Rutherford's atomic model's shortcomings, notably its inability to explain atomic stability. It then introduces the Bohr model, developed by Niels Bohr in 1913, which quantizes elec...This page examines Rutherford's atomic model's shortcomings, notably its inability to explain atomic stability. It then introduces the Bohr model, developed by Niels Bohr in 1913, which quantizes electron orbits within hydrogen, connecting quantum mechanics to atomic spectra. The page details Bohr's assumptions, the concept of the Bohr radius, and the Rydberg equation, alongside corrections for precise calculations.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/07%3A_Approximation_Methods/7.04%3A_Perturbation_Theory_Expresses_the_Solutions_in_Terms_of_Solved_ProblemsThis page examines perturbation theory in quantum mechanics, emphasizing its role in approximating energy and wavefunction changes from small Hamiltonian alterations. It focuses on first-order perturb...This page examines perturbation theory in quantum mechanics, emphasizing its role in approximating energy and wavefunction changes from small Hamiltonian alterations. It focuses on first-order perturbation, detailing energy shifts and wavefunction changes through specific examples, including harmonic oscillators and different potentials.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/14%3A_Nuclear_Magnetic_Resonance_Spectroscopy/14.07%3A_Spin-Spin_Coupling_Between_Chemically_Equivalent_Protons_is_Not_ObservedThis page discusses the concept of equivalent hydrogens in NMR spectroscopy, explaining that they do not affect each other's signals due to their identical chemical environments, resulting in singlet ...This page discusses the concept of equivalent hydrogens in NMR spectroscopy, explaining that they do not affect each other's signals due to their identical chemical environments, resulting in singlet signals. It details the Hamiltonian matrix for spin-$1/2$ particles, eigenstates, and transitions in NMR, while distinguishing between equivalent and non-equivalent nuclei.
- https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/05%3A_Electrons_in_Atoms/5.12%3A_Energy_LevelThis page explains how fireworks create colorful bursts of light through energy transitions of electrons in atoms. It outlines electron shells' roles in determining energy levels, and highlights that ...This page explains how fireworks create colorful bursts of light through energy transitions of electrons in atoms. It outlines electron shells' roles in determining energy levels, and highlights that valence electrons impact an atom's stability and reactivity. Reactive elements with incomplete valence shells, like fluorine and lithium, contrast with the stability of neon, which has a full outer shell.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/17%3A_Boltzmann_Factor_and_Partition_Functions/17.07%3A_Partition_Functions_of_Indistinguishable_Molecules_Must_Avoid_Over_Counting_StatesThis page discusses the partition function in statistical mechanics, comparing calculations for distinguishable and indistinguishable particles. Using a two-particle model, it shows that distinguishab...This page discusses the partition function in statistical mechanics, comparing calculations for distinguishable and indistinguishable particles. Using a two-particle model, it shows that distinguishable particles have four states, leading to a squared partition function. In contrast, indistinguishable particles have three states, requiring a modification to the partition function through a factor of N! to prevent overcounting.
- https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/05%3A_Electrons_in_Atoms/5.07%3A_Spectral_Lines_of_Atomic_HydrogenThis page discusses the evolution of scientific theory through automobile repairs and the Bohr model of the hydrogen atom. It highlights how energy changes in a hydrogen atom create spectral lines lin...This page discusses the evolution of scientific theory through automobile repairs and the Bohr model of the hydrogen atom. It highlights how energy changes in a hydrogen atom create spectral lines linked to electron transitions, successfully explaining the hydrogen emission spectrum but revealing limitations for multi-electron atoms. Key concepts include energy level quantization, the Balmer series, and the need for advancements in atomic theory.
