# The Bohr Model

## The Bohr modeldepicts atoms as small, positively charged nuclei surrounded by electrons in circular orbits.

#### 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 orbit the nucleus.

• The model states that the electrons can only orbit stably, without radiating, in certain 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, the electron's acceleration does not result in radiation and energy loss as required by classical electromagnetics.

#### Terms

• That which is emitted or sent out.

• To move along the path of a spiral or helix.

• Fuctuating; not constant.

#### Figures

1. ##### The Bohr Atom

The Rutherford–Bohr Model of the Hydrogen Atom.

2. ##### The Bohr Model of the Atom

We combine our new found knowledge of the nature of light with Bohr's atomic theory.

## The Bohr Model

In atomic physics, the Bohr model Figure 2 depicts the 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. Figure 1

The Bohr model was an improvement on the earlier cubic model (1902), the plum-pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911). Since the Bohr model is a quantum-physics-based modification of the Rutherford model, many sources combine the two—the Rutherford–Bohr model.

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.

The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics, and thus may be considered to be an obsolete scientific theory. Because of its simplicity and correct results for selected systems (see below for application), 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 full-blown quantum mechanics (1925) is often referred to as the old quantum theory.

Early "planetary" models 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.

In 1913, Bohr suggested that electrons could only have certain classical motions:

1. Electrons in atoms orbit the nucleus.
2. 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, the electron's acceleration does not result in radiation and energy loss as required by classical electromagnetics.
3. Electrons can only gain and 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.

Bohr's model is significan 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 three is not completely well defined for small orbits, Bohr could determine the energy spacing between levels using rule three and come to an exactly correct quantum rule: the angular momentum L is restricted to be an integer multiple of a fixed unit:

$L=n\frac { h }{ 2\pi } =n\hbar$

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 hydrogen-like atoms and ions.

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 νrot 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 En and En−k when k is much smaller than n. These jumps reproduce the frequency of the k-th harmonic of orbit n. For sufficiently large values of n (so-called 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 Bohr-Kramers-Slater 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 Glossary

angular momentum
The vector product that describes the rotary inertia of a system about an axis and is conserved in a closed system. For an isolated rigid body, it is a measure of the extent to which an object will continue to rotate in the absence of an applied torque.
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atom
the smallest possible amount of matter that still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons
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decay
To change by undergoing fission, by emitting radiation, or by capturing or losing one or more electrons.
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electromagnetic
Pertaining to electromagnetism.
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Radiation (quantized as photons) consisting of oscillating electric and magnetic fields oriented perpendicularly to each other, moving through space.
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electron
The subatomic particle having a negative charge and orbiting the nucleus; the flow of electrons in a conductor constitutes electricity.
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emission
That which is emitted or sent out.
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energy
a quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent
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frequency
The number of occurrences of a repeating event per unit of time.
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integer
An element of the infinite and numerable set {...,-3,-2,-1,0,1,2,3,...}.
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ion
An atom or group of atoms bearing an electrical charge, such as the sodium and chlorine atoms in a salt solution.
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momentum
(of a body in motion) the product of its mass and velocity.
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multiple
one of a set of the same thing; a duplicate
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nucleus
The massive, positively charged central part of an atom, made up of protons and neutrons.
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orbital
A specification of the energy and probability density of an electron at any point in an atom or molecule.
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period
a horizontal row in the periodic table, which signifies the total number of electron shells in an element's atom
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photoelectric effect
The emission of electrons from the surface of a material following the absorption of electromagnetic radiation
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photon
The quantum of light and other electromagnetic energy, regarded as a discrete particle having zero rest mass, no electric charge, and an indefinitely long lifetime. It is a gauge boson.
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proton
A positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element; the nucleus of the most common isotope of hydrogen; composed of two up quarks and a down quark
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quanta
discrete packets with energy stored inside
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quantization
the process of approximating a continuous signal by a set of discrete symbols or integer values
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quantum
the smallest possible, and therefore indivisible, unit of a given quantity or quantifiable phenomenon
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quantum number
One of certain integers or half-integers that specify the state of a quantum mechanical system (such as an electron in an atom).
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quantum theory
A theory developed in early 20th century, according to which nuclear and radiation phenomena can be explained by assuming that energy only occurs in discrete amounts called quanta.
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A line segment between any point on the circumference of a circle and its center/centre.
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spiral
To move along the path of a spiral or helix.
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state
The physical property of matter as solid, liquid, gas or plasma
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system
the part of the universe being studied, arbitrarily defined to any size desired
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unstable
Fuctuating; not constant.
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valence
the combining capacity of an atom, radical or functional group determined by the number of electrons that it will lose, gain, or share when it combines with other atoms, etc.
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valence shell
The outermost shell of electrons in an atom; these electrons take part in bonding with other atoms.