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Bohr's Atomic Model

10th Jan 2020 @ 8 min read

Physical Chemistry

The Bohr atomic model or Rutherford-Bohr model was an earlier atomic model developed by Niels Bohr in 1913. Bohr was a Danish Nobel laureate. The model supplanted the Rutherford model. It was an improvement to its predecessor Rutherford's atomic theory. The model was effective in explaining many limitations of its predecessors, especially the hydrogen spectrum. Bohr refined Rutherford's model with the help of the old quantum theory.

Niels Bohr
Niels Bohr (7 October 1885 – 18 November 1962)


Although the Bohr model is inaccurate, it is very helpful for understanding the quantized nature of subatomic particles, particularly electrons. Its postulates are described below.

  1. An atom consists of a dense, central nucleus with orbiting electrons. The electrons orbit around the nucleus in a circular path. The orbits are concentric having a fixed radius and fixed energy.
    Bohr's atomic model
    Bohr's model of atoms: the tiny nucleus with electrons revolving in the concentric orbits.
    The electrons are bounded to the nucleus by the electrostatic force between them.
  2. Unlike the earlier Rutherford model, the orbiting electrons do not continuously radiate energy. The orbits, aka stationary orbits, are stable and discrete with a fixed radius. The energy associated with each orbit is unchangeable. An electron can only revolve in these certain selected orbits. In other words, an electron cannot occupy any orbits other than selected orbits.
  3. When an electron jumps from one selected orbit to another, the energy is either absorbed or emitted. The difference in the energies of two orbits is hν.
    Energy difference between to orbits
    Here, Ei and Ef are the energies of the initial and final orbits, h is the Planck constant and its approximate value is 6.626 × 10−34 J s, and ν is the frequency of the radiation emitted or absorbed.
  4. If an electron jumps to a lower orbit, it emits energy.
    An electron emitting energy as it jumps to a lower orbit
    An electron emits energy as it jumps to a lower orbit. The negative sign with indicates energy is released.
    Similarly, when an electron absorbs energy, it moves to a higher orbit.
    An electron moves to a higher orbit when it absorbs energy.
    An electron moves to a higher orbit when it absorbs energy.
  5. The frequency of a radiation emitted or absorbed is given by the below formula.
    The frequency of a radiation
  6. The angular momentum of an electron is an integral multiple of h/2π. It means the angular momentum is quantized.
    1. me is the mass of the electron,
    2. v is the velocity of the electron,
    3. r is the radius of the orbit in which the electron is moving,
    4. n = 1, 2, 3… is called the principle quantum number.
    n = 1 is the lowest and innermost orbit, which is also closest to the nucleus. n = 2 is the second orbit, n = 3 is the second orbit, and so forth.

    Note: There are two ways in which we can represent the orbits of an atom: n = 1, 2, 3… and K, L, M… shell.

    The orbits of an atom
    The orbits of an atom

    Proof of mevr = nh

    De Broglie interpreted the electron as a standing wave. Thus, the circumference of the orbit of an electron should be an integral multiple of the wavelength.
    The circumference is an integral multiple of the wavelength
    Also, the De Broglie wavelength is given as:
    De Broglie wavelength
    Combining the above equations,
    Eliminating the wavelength
    The proof of the angular momentum
  7. Bohr also determined the radius and energy of orbits and the corresponding velocity of the electron. The radius of an electron orbiting in the n-th orbit is given as:
    The radius of the orbit
    1. Z is the atomic number of an atom,
    2. ke is the Coulomb constant (≈ 8.99 × 109 N m2 C−2),
    3. e is the charge on the electron (≈ 1.602 × 10−19 C),
    4. and me is the mass of the electron (≈ 9.109 × 10−31 kg).
    Substituting the value of the constants,
    The radius of the orbit as per Bohr's model
    In the above equation, the radius is in metres. Expressing the radius in the picometre,
    The radius in picometre

    Note: If notice, the radius of an atom is inversely proportional to the atomic number. Thus, the radius decreases with an increase in the atomic number—which is a general trend as we move from left to right in the modern periodic table.

    The energy of the orbit is followed by the below equation.
    Energy of the orbit

    Note: eV in the above expression stands for the electronvolt, a unit of energy. 1 eV ≈ 1.602 × 10−19 J.

    The velocity of the electron in the n-th orbit as per Bohr:
    The velocity of the electron according to Bohr

    Derivation of the above formulae

    In an atom, the negatively charged electrons circles around the positively charged nucleus. They are held by the electrostatic force.
    Centripetal force equals electrostatic force
    The electron experiences a centripetal force when it orbits around the nucleus.
    In circular motion, the revolving object is continuously acted by centripetal force, which in an atom is electrostatic force. Thus, centripetal force equals electrostatic force.
    Centripetal force
    Cancelling the radius
    From a Bohr's postulate,
    Angular momentum
    velocity in terms of radius
    Substituting v,
    Eliminating the velocity
    The radius of the orbit according to Bohr
    The velocity of the electron:
    The velocity of electron
    The energy of an electron:
    Kinetic energy of an electron
    Energy of the electron according to Bohr's model

Bohr's model and hydrogen atom

Bohr's model holds good for the hydrogen atom. Hydrogen atom is the simplest atom with one proton and one electron. The model is also applicable to ions similar to hydrogen, e.g. He+, Li2+, Be3+, which have only one electron.

The radius of the hydrogen atom (Z = 1):

The radius of hydrogen as per Bohr's model

When n = 1, r1 = 52.9 pm. This is the first stationary orbit, and its value is also called the Bohr radius. For n = 2, 3…, r = 211.6 pm, 476.1 pm…

The velocity of the hydrogen atom:

The velocity of the hydrogen atom

For n = 1, 2, 3…, v = 2.187 m s−1, 1.094 m s−1, 0.547 m s−1… Therefore, the velocity of an electron in the first orbit is the highest.

The energy of the hydrogen atom:

The energy of the hydrogen atom

The energy of the first orbit (n = 1) is −13.6 eV. For n = 2, 3…, −3.40 eV, −1.51 eV… Although the energy increases with n, its absolute value decreases.

An electron in the hydrogen atom is mostly found in its ground state, the lowest energy state. The electron moves to a higher energy state, an excited state, when it absorbs energy.

The energy of the electron is negative; it indicates we have to supply energy to release an electron from an orbit. When n approaches infinity, the energy tends to zero. Hence, no energy is required and it is called the free electron.

Energy tends zero at larger orbits

As explained earlier, an electron can jump from an orbit to another. In this process, it emits energy if it descends or absorbs energy if it ascends. This is similar to a staircase. Every stair represents a stationary orbit. When an electron climbs down stairs, its energy is reduced and the difference in the energies is emitted. Similarly, when external energy is supplied, it absorbs and climbs up stairs. The figure below depicts the same.

Hydrogen atom with its energy level
Hydrogen atom with its energy levels

We can even calculate the difference in the energy levels in a transition of an electron.

Let Ei and Ef be the energies in the initial and final states.

Initial and final Energies

The difference between both is ∆E.

Energy difference between the initial and final stages

We also know, ∆E = .

Substituting ∆E

The frequency of an electromagnetic radiation is The frequency of the radiation is the speed of light divided by its wavelength.. Here, c and λ are the speed of light (≈ 3 × 108 m s−1) and the wavelength.

Substituting the frequency Rearranging the above equation Wavenumber

Here, ν̄ is the wavenumber, which is equal to the reciprocal of the wavelength, and RH is the Rydberg constant of hydrogen and its value is given below.

Rydberg constant of hydrogen = 1.09678 × 10^7 m^-1

Hydrogen spectra

One of the achievements of Bohr's model was that it could explain the hydrogen spectra.

When a white light is projected on a sample of hydrogen, electrons absorb the radiation and transit to an excited state. Since electrons can only absorb the light of certain wavelengths, most of the light remains unabsorbed. The outcoming light will be deficient in these absorbed wavelengths. This can be seen in the absorption spectrum of hydrogen (see the figure below).

Hydrogen absorption spectrum
Hydrogen absorption spectrum: the missing wavelengths in the above figure are absorbed by the hydrogen sample.

The emission spectrum of hydrogen is similar to that of the absorption spectrum. Here, the light coming from an excited sample of hydrogen is used. The wavelengths emitted are the same in the absorption spectrum. See the figure below; it is self-explanatory.

Hydrogen emission spectrum
Hydrogen emission spectrum: The bright lines in the spectrum are the wavelengths emitted by the excited sample of hydrogen.

None of the earlier models to Bohr's was able to explain these spectral lines. In fact, we can even calculate each wavelength using the equation below.

hydrogen spectral lines formula

Let's say an electron jumps from n = 2 to n = 3.

wavelength when ni=2 and nf=3

The wavelength from the above equation is 656.47 nm. The value matches that in the spectral diagrams. In the same manner, we can determine the wavelengths for ni = 2 and nf = 4, 5, 6, 7…∞; the table below represents all of them.

The series of wavelengths, aka the Balmer series
ninfλ (nm)Colour
28389.01near UV
29383.65near UV
2364.70In UV

The above series of the wavelengths, aka the Balmer series, reciprocates the hydrogen spectral lines.


Although the Bohr model was successful in explaining the hydrogen spectra, it had several limitations and was replaced by the quantum mechanic model.

  1. The model stood strong in explaining light spectra of lighter atoms similar to hydrogen. It could not explain the spectral lines for heavier atoms.
  2. It failed to mention the dual nature (wave-particle duality) of the electron.
  3. The model contradicted the Heisenberg uncertainty principle. According to the principle, the position and momentum of a particle cannot be determined simultaneously. However, the model defined the position (orbits) and momentum of the electron at the same time.
  4. It failed to explain the splitting of spectral lines in the presence of an external magnetic field (the Zeeman effect) and an external electric field (the Stark effect).
  5. As per the Bohr's model, the angular momentum of the electron in the ground state of a hydrogen atom is equal to the reduced Planck constant (h), which it is untrue. The modern quantum theory says it is zero.
  6. The model was also inefficient to account several subtle details in spectra like fine and hyperfine structure, doublets and triplets.

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