Bohr's Atomic Model

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.

Postulates

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.

   

   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.
   
   Here, E_i and E_f 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.

   

   Similarly, when an electron absorbs energy, it moves to a higher orbit.

   

5. The frequency of a radiation emitted or absorbed is given by the below formula.
   
6. The angular momentum of an electron is an integral multiple of . It means the angular momentum is quantized.
   
   Where:
   1. m_e 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.

   

   Proof of m_evr = ^nh⁄_2π

   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.

   

   Also, the De Broglie wavelength is given as:

   

   Combining the above equations,

   

   
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:
   
   Here,
   1. Z is the atomic number of an atom,
   2. k_e 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 m_e is the mass of the electron (≈ 9.109 × 10⁻³¹ kg).
   Substituting the value of the constants,
   
   In the above equation, the radius is in metres. Expressing the radius in the 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.
   

   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:
   

   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.

   

   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.

   

   

   From a Bohr's postulate,

   

   

   Substituting v,

   

   

   The velocity of the electron:

   

   The energy of an electron:

   

   

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):

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:

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 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.

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.

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

Let E_i and E_f be the energies in the initial and final states.

The difference between both is ∆E.

We also know, ∆E = hν.

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

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

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).

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.

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.

Let's say an electron jumps from n = 2 to n = 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 n_i = 2 and n_f = 4, 5, 6, 7…∞; the table below represents all of them.

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

Drawbacks

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⁄_2π), 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.

Associated articles

• Rutherford's atomic model
• Rutherford's gold foil Experiment
• Plum pudding atomic model by J. J. Thomson
• J. J. Thomson's cathode ray tube experiments