Bohr Model of a Hydrogen Atom – its Postulates, Derivation of the Relations

The Bohr model of a hydrogen atom was proposed by scientist Neils Bohr in the year 1913. According to the Bohr model formula of hydrogen, hydrogen atoms have a positively charged nucleus, which possesses protons, and neutrons. They are surrounded by negatively charged electrons and are seen to orbit around the nucleus in atomic shells. There are strong electrostatic forces of attraction between this positively charged nucleus and the negatively charged electron. The first atomic model was successful in explaining the radiation spectra of the hydrogen atoms. Neil Bohr was able to fill up the gaps that were created by Rutherford’s model.

Bohr model of a hydrogen

In 1915, Niels Bohr proposed Bohr’s Atomic Model. This model was a modification of Rutherford’s atomic model. Bohr’s model of an atom is also known as the Rutherford Bohr model formula.

According to Rutherford’s model, a nucleus is positively charged and is surrounded by negatively charged electrons. Bohr modified this model by adding that the electrons travel in fixed circular orbits around the nucleus. No electron travels between these fixed orbits. Each orbital or shell has a definite energy.

What is Bohr’s Atomic Model?

Bohr’s model of an atom is similar to the planetary model. In Bohr’s model, electrons move in fixed circular orbits around a positively charged nucleus. The energy associated with each orbit is fixed. Each circular orbit has a fixed distance from the nucleus. 

Bohr’s model of an atom explains the electron’s properties in terms of allowed values. Bohr’s model can explain the absorption and emission of radiation when an electron makes a transition between different energy levels. In 1922, Bohr was awarded the Nobel Prize in physics for his work.

Postulates of Bohr’s hydrogen atom:

Bohr combined both the early Quantum theory and the classical theory of hydrogen atoms and suggested three postulates. Let us discuss in detail about these postulates.

• Postulate 1: In this postulate, he proposed that electrons revolve in stable orbits without emitting any radiant energy. He also stated that atoms can exist in stable states and have definite total energy.
• Postulate 2: According to Bohr’s second postulate, the orbits have full stability and the electrons revolve around the nucleus in orbits. He gave the theory of angular momentum of revolution. It is an integral multiple of h/2p where the Planck’s constant and h is equal to 6.6×10^-34J-s.
• Postulate 3: Bohr proposed early Quantum concepts into the atomic theory. In this theory, the transition of electrons happens in a non-radiating orbit to a lower energy level. In this transition process, photons are emitted. The energy of the photos is equal to the difference between the other two states.

The equation is as follows:

                         hv= Ei- Ef

where, h is Planck’s constant, v is the frequency of the emitted photon, Ei is the energy of the initial state, and Ef stands for the energy of the ultimate state. Also, Ei is always more than Ef.

Formulas of Bohr’s postulates

To determine the relationship between the number of the orbit and electrons’ angular moment, the following Bohr model formula is used: 

mvr = nh/2𝜋

where m is the mass of an electron, v is the velocity, r is the radius of the orbit, h is Planck’s constant, and n is the number of the orbit. 

For defining the energy emission or absorption of the electrons, Bohr’s formula for energy levels: 

E1 – E2 = hf

Or, En = – (2𝜋2me4z2k2/n2h2)

Derivation of Relations:

After the first proposal of the planetary model of the hydrogen atom, Neil Bohr made an assumption. And it’s the quantization of the atomic structure. According to Bohr, electrons make orbits around the nucleus with a fixed radius.

Therefore the value of atomic radius can be calculated by the following equation:

r(n) = n2× r(1)

Here, n is any positive integer, r(1) is the smallest radius for the hydrogen atom. This smallest radius is termed Bohr’s radius. The value of Bohr’s radius is 0.529×10-10m.

The energy calculation of an electron at the nth level of hydrogen is formed by considering the electrons in circular orbits. The equation is as follows:

E(n)= 1/n2 × 13.6eV

Here, the lowest possible energy of a hydrogen electron is 13.6 eV.

Important note: the obtained energy from this equation always gives a negative value. And the highest negative value is obtained when n is equal to 1. This is so because the energy of an electron relative to that energy of an electron in orbit is separated from its nucleus when n=infinity. And in this case, the energy is 0 eV. And when an electron is far away from its nucleus its energy is less than that which is in a fixed orbit around the nucleus so the energy of an electron in orbit is always negative.

Conclusion:

The Bohr model of a hydrogen atom is indeed completely different from that of the modern quantum mechanical model but their idea is the same. There are some limitations of Bohr’s model of the hydrogen atom. Let us learn in detail about these limitations.

• This model does not work for that complex atoms
• Bohr’s theory of hydrogen spectrum failed in explaining why spectral lines are more intense than other spectral lines.
• Bohr’s idea of electrons in orbits is contradicted by Heinberg’s uncertainty principle with unknown radius and velocity
• It is also unsuccessful in explaining why some spectral lines split into multiples in the presence of a magnetic field.