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The Bohr Model
The Bohr model depicts atoms as small, positively charged nuclei surrounded by electrons in circular orbits.
Learning Objectives

Describe the limitations of the Bohr model that arise from its use of Maxwell's theory

Explain how the Bohr model of the atom marked an improvement over earlier models
Key Points
 The model's success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen.
 The model states that electrons in atoms move in circular orbits around a central nucleus and can only orbit stably in certain fixed circular orbits at a discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels.
 In these stable orbits, an electron's acceleration does not result in radiation and energy loss as required by classical electromagnetic theory.
Terms

unstable
For an electron orbiting the nucleus, according to classical mechanics, it would mean an orbit of decreasing radius and approaching the nucleus in a spiral trajectory.

emission
Act of releasing or giving away, energy in the case of the electron.

correspondence principle
States that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum number.
Full Text
In atomic physics, the Bohr model depicts an atom as a small, positively charged nucleus surrounded by electrons. These electrons travel in circular orbits around the nucleus—similar in structure to the solar system, except electrostatic forces rather than gravity provide attraction.
The Bohr atom
The Rutherford–Bohr model of the hydrogen atom. In this view, electron orbits around the nucleus resemble that of planets around the sun in the solar system.
Development of the Bohr Model
The Bohr model was an improvement on the earlier cubic model (1902), the plumpudding model (1904), the Saturnian model (1904), and the Rutherford model (1911). Since the Bohr model is a quantumphysicsbased modification of the Rutherford model, many sources combine the two: the Rutherford–Bohr model.
Although it challenged the knowledge of classical physics, the model's success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen. While the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, it also provided a justification for its empirical results in terms of fundamental physical constants.
Although revolutionary at the time, the Bohr model is a relatively primitive model of the hydrogen atom compared to the valence shell atom. As an initial hypothesis, it was derived as a firstorder approximation to describe the hydrogen atom. Due to its simplicity and correct results for selected systems, the Bohr model is still commonly taught to introduce students to quantum mechanics. A related model, proposed by Arthur Erich Haas in 1910, was rejected. The quantum theory from the period between Planck's discovery of the quantum (1900) and the advent of a fullblown quantum mechanics (1925) is often referred to as the old quantum theory.
Early planetary models of the atom suffered from a flaw: they had electrons spinning in orbit around a nucleus—a charged particle in an electric field. There was no accounting for the fact that the electron would spiral into the nucleus. In terms of electron emission, this would represent a continuum of frequencies being emitted since, as the electron moved closer to the nucleus, it would move faster and would emit a different frequency than those experimentally observed. These planetary models ultimately predicted all atoms to be unstable due to the orbital decay. The Bohr theory solved this problem and correctly explained the experimentally obtained Rydberg formula for emission lines.
Properties of Electrons under the Bohr Model
In 1913, Bohr suggested that electrons could only have certain classical motions:
 Electrons in atoms orbit the nucleus.
 The electrons can only orbit stably, without radiating, in certain orbits (called by Bohr the "stationary orbits") at a certain discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, an electron's acceleration does not result in radiation and energy loss as required by classical electromagnetic theory.
 Electrons can only gain or lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency (ν) determined by the energy difference of the levels according to the Planck relation.
Behavior of Electrons: Part 3, The Bohr Model of the Atom  YouTube
We combine our new found knowledge of the nature of light with Bohr's atomic theory.
Bohr's model is significant because the laws of classical mechanics apply to the motion of the electron about the nucleus only when restricted by a quantum rule. Although Rule 3 is not completely well defined for small orbits, Bohr could determine the energy spacing between levels using Rule 3 and come to an exactly correct quantum rule—the angular momentum L is restricted to be an integer multiple of a fixed unit:
where n = 1, 2, 3, ... is called the principal quantum number and ħ = h/2π. The lowest value of n is 1; this gives a smallest possible orbital radius of 0.0529 nm, known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogenlike atoms and ions.
The Correspondence Principle
Like Einstein's theory of the photoelectric effect, Bohr's formula assumes that during a quantum jump, a discrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels. Bohr did not believe in the existence of photons.
According to the Maxwell theory, the frequency (ν) of classical radiation is equal to the rotation frequency (ν_{rot}) of the electron in its orbit, with harmonics at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels E_{n} and E_{n−k} when k is much smaller than n. These jumps reproduce the frequency of the kth harmonic of orbit n. For sufficiently large values of n (socalled Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency so that the classical orbital frequency is not ambiguous. But for small n (or large k), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the correspondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
The BohrKramersSlater theory (BKS theory) is a failed attempt to extend the Bohr model, which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average.
Key Term Reference
 angular momentum
 Appears in these related concepts: Angular Quantities as Vectors, Angular vs. Linear Quantities, and Conservation of Angular Momentum
 atom
 Appears in these related concepts: The Law of Multiple Proportions, Stable Isotopes, and John Dalton and Atomic Theory
 decay
 Appears in these related concepts: Radioactive Decay Series: Introduction, Models Using Differential Equations, and Discovery of Radioactivity
 electromagnetic radiation
 Appears in these related concepts: Scattering of Light by the Atmosphere, Other Forms of Energy, and Gamma Decay
 electron
 Appears in these related concepts: Periods 1 through 3, The Pauli Exclusion Principle, and Overview of Atomic Structure
 energy
 Appears in these related concepts: Energy Transportation, Surface Tension, and Introduction to Work and Energy
 integer
 Appears in these related concepts: Exact Numbers, Scientific Notation, and Finding a Specific Term
 ion
 Appears in these related concepts: Solutions and Heats of Hydration, Ions, and Ions and Ionic Bonds
 momentum
 Appears in these related concepts: Conservation of Energy and Momentum, Elastic Collisions in One Dimension, and Elastic Collisions in Multiple Dimensions
 nucleus
 Appears in these related concepts: Clusters of Neuronal Cell Bodies, Early Models of the Atom, and Electric Charge in the Atom
 orbital
 Appears in these related concepts: Periodic Table Position and Electron Configuration, Color, and Electron Orbitals
 period
 Appears in these related concepts: The Periodic Table, Number of Periods, and Atomic Size
 photoelectric effect
 Appears in these related concepts: Diffraction Revisited, Energy, Mass, and Momentum of Photon, and Basic Assumptions of the Bohr Model
 photon
 Appears in these related concepts: Hydrogen Spectra, XRays and the Compton Effect, and Energy and Momentum
 proton
 Appears in these related concepts: Overview of the AcidBase Properties of Salt, Balancing Nuclear Equations, and Development of the Periodic Table
 quanta
 Appears in these related concepts: The Uncertainty Principle, Particle in a Box, and Distribution of Molecular Speeds and Collision Frequency
 quantization
 Appears in these related concepts: General Rules for Assigning Electrons to Atomic Orbitals, Bohr Orbits, and Wave Nature of Matter Causes Quantization
 quantum
 Appears in these related concepts: Linear Combination of Atomic Orbitals (LCAO), The de Broglie Wavelength, and The Bottom of the Periodic Table
 quantum number
 Appears in these related concepts: Quantum Numbers, Wave Equation for the Hydrogen Atom, and Diamagnetism and Paramagnetism
 quantum theory
 Appears in these related concepts: Atomic Radius and Planck's Quantum Theory
 system
 Appears in these related concepts: Screening, Definition of Management, and Comparison of Enthalpy to Internal Energy
 valence
 Appears in these related concepts: General Trends in Chemical Properties, Expectancy Theory, and Expectancy Theory
 valence shell
 Appears in these related concepts: Electron Configurations, The Shielding Effect and Effective Nuclear Charge, and Multielectron Atoms
Sources
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Source: Boundless. “The Bohr Model.” Boundless Chemistry. Boundless, 21 Jul. 2015. Retrieved 28 Aug. 2015 from https://www.boundless.com/chemistry/textbooks/boundlesschemistrytextbook/introductiontoquantumtheory7/bohrstheory64/thebohrmodel2996579/