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30.3 Bohr’s Theory of the Hydrogen Atom

3 min read•june 18, 2024

of the hydrogen atom revolutionized our understanding of atomic structure. It explained the discrete nature of atomic spectra and introduced the concept of , paving the way for modern .

The model proposed electrons orbiting the in specific energy states, with transitions between states causing photon emission or absorption. While it had limitations, Bohr's theory successfully predicted hydrogen's spectral lines and laid crucial groundwork for future atomic models.

Bohr's Theory of the Hydrogen Atom

Significance of atomic spectra

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  • Unique patterns of light emitted or absorbed by atoms
    • Hydrogen's consists of distinct wavelengths ( in ultraviolet, in visible, in infrared)
    • is the inverse of the emission spectrum
  • Classical physics could not explain the discrete nature of atomic spectra
    • Expected a continuous spectrum based on classical electromagnetic theory
  • Bohr's theory aimed to provide a theoretical explanation for the observed atomic spectra

Key principles of Bohr's model

  • Electrons orbit the nucleus in circular paths called or energy levels
    • Each state has a specific radius and energy
    • energies are quantized, meaning they can only have certain discrete values
  • Electrons transition between states by absorbing or emitting
    • Absorbing a photon causes an electron to jump to a higher energy level
    • Emitting a photon causes an electron to drop to a lower energy level
  • Energy of the absorbed or emitted photon equals the difference in energy between the two states
    • ΔE=hf\Delta E = hf, where hh is and ff is the photon
  • of an electron in a stationary state is quantized
    • mvr=nh2πmvr = n\frac{h}{2\pi}, where nn is an integer, mm is electron mass, vv is velocity, and rr is orbit radius
  • The force of attraction between the electron and nucleus is described by

Energy-level diagrams for hydrogen

  • Show the relative energies of an atom's stationary states
    • Each horizontal line represents a specific energy level
    • Lowest energy level () is usually denoted as n=1n=1
    • Higher energy levels have increasing nn values (n=2,3,4,...n=2, 3, 4, ...)
  • Arrows between levels represent transitions caused by photon absorption or emission
    • Upward arrows indicate absorption, downward arrows indicate emission
  • of the absorbed or emitted photon is related to the energy difference between levels
    • 1λ=R(1n12−1n22)\frac{1}{\lambda} = R(\frac{1}{n_1^2} - \frac{1}{n_2^2}), where RR is the , and n1n_1 and n2n_2 are the initial and final levels
  • The energy required to remove an electron from the atom completely is called the

Bohr's model vs earlier theories

  • Improved upon the
    • Rutherford proposed a "solar system" model with electrons orbiting a positively charged nucleus
    • Bohr introduced the concept of quantized energy levels to explain atomic spectra
  • Differed from Thomson's "plum pudding" model
    • Thomson suggested electrons were embedded in a positively charged "pudding"
    • Bohr's model had electrons orbiting a dense, positively charged nucleus

Successes and limitations of Bohr's theory

  • Successes:
    1. Explained the discrete nature of atomic spectra for hydrogen
    2. Accurately predicted the wavelengths of light in hydrogen's emission spectrum
    3. Introduced the concept of quantized energy levels in atoms
  • Limitations:
    1. Could not accurately explain the spectra of atoms with more than one electron
    2. Did not account for the wave-like properties of electrons
    3. Unable to explain the fine structure and hyperfine structure of spectral lines
  • Despite limitations, laid the groundwork for the development of

Quantum mechanical developments

  • : Introduced the concept of wave-particle duality for electrons
  • : Provided a more accurate description of electron behavior in atoms
  • Atomic number: Determines the number of protons in the nucleus and electrons in a neutral atom

Key Terms to Review (41)

Absorption Spectrum: An absorption spectrum is the unique pattern of dark lines or bands that appear in the continuous spectrum of light that has passed through a gaseous medium. This spectrum is characteristic of the elements present in the gas and provides a fingerprint for identifying those elements.
Angular momentum: Angular momentum is the rotational analog of linear momentum, representing the quantity of rotation of an object. It is a vector quantity given by the product of an object's moment of inertia and its angular velocity.
Angular Momentum: Angular momentum is a measure of the rotational motion of an object around a fixed axis. It describes the object's tendency to continue rotating and the amount of torque required to change its rotational state. This concept is fundamental in understanding the dynamics of rotating systems and is crucial in various areas of physics, from the motion of satellites to the behavior of subatomic particles.
Antielectron: An antielectron, also known as a positron, is the antimatter counterpart of an electron. It has the same mass as an electron but carries a positive charge.
Balmer series: The Balmer series refers to a set of spectral lines that correspond to the transitions of an electron in a hydrogen atom from higher energy levels down to the second energy level. These transitions release specific wavelengths of light, which are visible and contribute to the color spectrum seen in hydrogen gas. The series highlights the quantized nature of energy levels in atoms and is a direct consequence of the principles established by early quantum theory.
Beat frequency: Beat frequency is the frequency at which two waves of slightly different frequencies interfere with each other, resulting in a modulation pattern perceived as a periodic variation in amplitude. It is calculated as the absolute difference between the frequencies of the two interfering waves.
Bohr radius: The Bohr radius is the average distance between the proton and the electron in a hydrogen atom in its ground state. It is a fundamental physical constant denoted by $a_0$.
Bohr's Theory: Bohr's Theory is a model of the atom that was developed by Danish physicist Niels Bohr in 1913. It provides a detailed explanation of the structure and behavior of the hydrogen atom, and laid the foundation for our understanding of atomic physics.
Coulomb's Law: Coulomb's law is a fundamental principle in electrostatics that describes the force of interaction between two stationary electric charges. It establishes a mathematical relationship between the magnitude of the electrostatic force, the charges involved, and the distance between them.
De Broglie wavelength: The de Broglie wavelength is the wavelength associated with a particle and is inversely proportional to its momentum. It highlights the wave-particle duality of matter.
Double-slit interference: Double-slit interference is a phenomenon that occurs when coherent light passes through two closely spaced slits, creating an interference pattern of bright and dark fringes on a screen. This experiment demonstrates the wave nature of light.
Electron: An electron is a fundamental subatomic particle that carries a negative electric charge and is found in all atoms, playing a crucial role in various physical and chemical phenomena. Electrons are responsible for the flow of electric current, the formation of chemical bonds, and the behavior of matter at the atomic and molecular levels. The concept of the electron is central to understanding topics such as static electricity, electric fields, magnetic fields, the photoelectric effect, quantum mechanics, and the structure of atoms. Electrons are the building blocks of matter and are essential for understanding the fundamental nature of the universe.
Emission spectrum: An emission spectrum is a spectrum of the electromagnetic radiation emitted by a substance when its atoms or molecules are excited and then return to lower energy states. This process results in the release of light at specific wavelengths, which appear as distinct lines on the spectrum, unique to each element. The emission spectrum is crucial for identifying elements and understanding atomic structure.
Energies of hydrogen-like atoms: Energies of hydrogen-like atoms are quantized energy levels derived from Bohr's theory, which describes the behavior of electrons in atoms with a single electron. These energy levels depend on the principal quantum number and the atomic number.
Energy Levels: Energy levels refer to the discrete, quantized states that electrons can occupy within an atom or molecule. These energy levels represent the specific amounts of energy an electron can have, and transitions between these levels are the basis for many important phenomena in quantum mechanics and atomic physics.
Energy-level diagram: An energy-level diagram is a graphical representation showing the allowed energy states (or levels) of an electron in an atom. These diagrams illustrate transitions between different energy levels, often using arrows to indicate the absorption or emission of photons.
Energy-Level Diagrams: Energy-level diagrams are visual representations that depict the discrete energy levels of an atom or molecule. These diagrams illustrate the allowed energy states that an electron can occupy within an atom's or molecule's structure, providing insights into the behavior and properties of these fundamental particles.
Ernest Rutherford: Ernest Rutherford was a physicist known as the father of nuclear physics, recognized for his pioneering work in the early 20th century on the structure of the atom. His groundbreaking gold foil experiment led to the discovery of the atomic nucleus, transforming our understanding of atomic structure and laying the groundwork for future theories about atomic particles and their interactions.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event per unit of time. It is a crucial parameter in various areas of study, including radiation, oscillations, waves, sound, and electromagnetic phenomena.
Ground state: The ground state is the lowest energy level of an atom or a quantum system, where the particles within are in their most stable configuration. In this state, electrons occupy the lowest available energy levels, and no external energy has been added to excite them to higher levels. Understanding the ground state is essential for grasping how energy is quantized and how photons are emitted or absorbed in various processes.
Hydrogen-spectrum wavelengths: Hydrogen-spectrum wavelengths are specific wavelengths of light emitted or absorbed by hydrogen atoms as electrons transition between energy levels. These wavelengths form discrete lines in the spectrum, known as spectral lines.
Ionization Energy: Ionization energy is the amount of energy required to remove an electron from an atom or molecule, forming a positive ion. It is a fundamental property that determines the chemical reactivity and stability of elements and compounds.
J.J. Thomson: J.J. Thomson was a British physicist best known for discovering the electron and proposing the plum pudding model of the atom. His work fundamentally changed the understanding of atomic structure and paved the way for future developments in atomic theory, emphasizing the presence of smaller particles within atoms.
Lyman series: The Lyman series is a set of spectral lines that represent the transitions of an electron in a hydrogen atom from higher energy levels down to the lowest energy level, n=1. This series is part of the hydrogen emission spectrum and occurs in the ultraviolet region of the electromagnetic spectrum. The quantized energy levels of the hydrogen atom dictate these transitions, resulting in the emission of light at specific wavelengths.
Niels Bohr: Niels Bohr was a Danish physicist known for his foundational contributions to understanding atomic structure and quantum theory, particularly through his model of the hydrogen atom. His work introduced the idea that electrons move in specific orbits around the nucleus and that energy is quantized, which means electrons can only exist in certain energy levels. This concept revolutionized the way scientists viewed atomic behavior and laid the groundwork for modern quantum mechanics.
Nucleus: The nucleus is the central and most important part of an atom, containing the protons and neutrons that make up the atom's core. It is the defining feature of an atom and plays a crucial role in the structure and behavior of matter at the most fundamental level.
Orbital: An orbital is a mathematical function that describes the wave-like behavior of electrons in an atom, representing the probability distribution of an electron's position around the nucleus. These orbitals can have different shapes and orientations, which are essential for understanding how atoms bond and interact with each other.
Paschen series: The Paschen series refers to a set of spectral lines in the infrared region produced when an electron transitions from a higher energy level (n ≥ 4) to the n = 3 energy level in a hydrogen atom. This series is a crucial aspect of Bohr’s theory of the hydrogen atom, which describes how electrons orbit the nucleus and can move between specific energy levels, resulting in the emission or absorption of light.
Photons: Photons are the fundamental particles that make up light and other forms of electromagnetic radiation. They are the quanta, or discrete packets, of energy that exhibit both particle-like and wave-like properties, and are central to understanding the behavior of light and its interactions with matter.
Planck's constant: Planck's constant is a fundamental physical constant that represents the smallest possible change in energy or action. It is a crucial parameter in quantum mechanics and is denoted by the symbol 'h'. Planck's constant establishes the relationship between the energy of a photon and its frequency, and it is a key factor in understanding the quantization of energy and the wave-particle duality of matter and energy.
Quantized: Quantized refers to the concept that certain physical quantities, such as energy levels, exist in discrete values rather than a continuous range. This idea is fundamental in understanding atomic and subatomic systems, where particles like electrons can only occupy specific energy states. It highlights how energy is absorbed or emitted in fixed amounts, often referred to as quanta.
Quantum mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of particles at atomic and subatomic scales. It explains phenomena that cannot be accounted for by classical physics.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy on the atomic and subatomic scale. It is a powerful framework for understanding the properties and interactions of particles at the quantum level, which are often counterintuitive and defy classical physics.
Radius of a nucleus: The radius of a nucleus is the distance from the center of the nucleus to its outer edge. It is typically measured in femtometers (fm), where 1 fm = $10^{-15}$ meters.
Rutherford model: The Rutherford model is a foundational concept in atomic physics that proposes the structure of the atom as having a small, dense nucleus containing positively charged protons, surrounded by negatively charged electrons in orbits. This model emerged from Ernest Rutherford's gold foil experiment, which demonstrated that most of an atom's mass is concentrated in the nucleus, leading to significant advancements in understanding atomic structure and paving the way for later theories like Bohr’s theory of the hydrogen atom.
Rydberg constant: The Rydberg constant is a physical constant related to the electromagnetic spectra of atoms. It represents the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from the hydrogen atom.
Rydberg Constant: The Rydberg constant is a fundamental physical constant that describes the wavelengths of the spectral lines emitted by the hydrogen atom. It is a crucial parameter in Bohr's theory of the hydrogen atom and is named after the Swedish physicist Johannes Rydberg, who first derived the formula for the wavelengths of the hydrogen spectral series.
Schrödinger Equation: The Schrödinger equation is a fundamental equation in quantum mechanics that describes the wave function of a particle and how it evolves over time. It is a central concept that connects the particle-wave duality and the quantization of energy, and is essential for understanding the behavior of quantum systems, including the structure of atoms and the tunneling phenomenon.
Stationary states: Stationary states refer to specific quantum states of a system, particularly an atom, where the system's properties remain constant over time. In the context of an atom, these states correspond to fixed energy levels where an electron can exist without radiating energy, meaning they are stable configurations that do not change unless acted upon by an external force.
Thomson model: The Thomson model, also known as the plum pudding model, is an early atomic theory proposed by J.J. Thomson in 1904, which describes the atom as a sphere of positive charge with negatively charged electrons embedded within it. This model attempted to explain the structure of atoms and their electrical properties, paving the way for future theories about atomic structure.
Wavelength: Wavelength is a fundamental characteristic of waves, representing the distance between consecutive peaks or troughs in a wave. It is a crucial parameter that describes the spatial extent of a wave and is closely related to other wave properties such as frequency and speed.
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30.4 X Rays: Atomic Origins and Applications