Bohr Model of a Hydrogen Atom

Introduction

The emission and absorption spectra of atomic hydrogen and hydrogen-like ions with low atomic numbers are explained by Bohr’s model of the hydrogen atom. It was the first model to use a quantum number to describe atomic states and to postulate the quantisation of electron orbits within the atom. The development of quantum mechanics, which deals with many-electron atoms, was aided by Bohr’s model.

How Do You Derive The Bohr Equation?

• The photons’ energies are quantised, and their energy is calculated as the change in energy of the electron as it moves from one orbit to the next. 

In equation form, this is written as E = hf = Ei × Ef. 

•  The orbits of the electrons are quantised in Bohr’s modified planetary model of the atom.

Using Bohr’s model, the energies for an electron in the shell are calculated:

E(n) = –  1n2eV

The hydrogen spectrum was explained by Bohr in terms of electrons absorbing and emitting photons to alter energy levels, where the photon energy is measured in eV.

hν=ΔE= – 13.6 (1nlow2 – 1nhigh2) eV

Note: Bohr’s model doesn’t work for systems with more than one electron.

Postulates of Bohr’s model of an atom

• Electrons (negatively charged) in an atom orbit around the positively charged nucleus in a specific circular path.
• These circular orbits are known as orbital shells because each orbit or shell has defined energy.
• The quantum number is an integer (n=1, 2, 3,…) that represents the energy levels. The lowest energy level is n=1 in this range of quantum numbers, which starts from the nucleus side. K, L, M, N…. shells are given to the orbits n=1, 2, 3, 4…, and an electron is considered to be in the ground state when it reaches the lowest energy level.
• An electron in an atom gains energy to move from a lower energy level to a higher energy level, and an electron loses energy to move from a higher energy level to a lower energy level.

Bohr introduced three postulates of Bohr’s model to overcome these two difficulties:

1. The negative electron orbits the positive nucleus (proton) in a circular orbit. The nucleus is the centre of all electron-orbits. An electron bonded to the nucleus does not have access to all possible orbits.
2. The electron orbits that are permitted satisfy the first quantisation condition: The electron’s angular momentum can only take discrete values in the nth orbit:

Ln = nh, where n = 1,2,3

The angular momentum of the electron is quantised according to this assumption. The first quantization condition can be written formally as:

mcvnrn = nh

and is denoted by the radius of the nth orbit and the electron’s speed in it, respectively.

3. Transitions from one orbit with En energy to another orbit with Em energy are permitted for an electron. When a photon is absorbed by an atom, the electron moves to a higher-energy orbit. When a photon is emitted by an atom, the electron moves to a lower-energy orbit. Electron transitions with simultaneous photon absorption or emission occur in a split second. The second quantisation criterion is satisfied by the allowed electron transitions:

hf =  En-Em
Where h is the energy of a photon with frequency f that is either emitted or absorbed. 

• According to the second quantisation condition, the change in energy of an electron in the hydrogen atom is quantised.
• These three postulates of the early quantum theory of the hydrogen atom allow us to derive the Rydberg formula and the value of the Rydberg constant. It also helps understand the important properties of the hydrogen atom, like its energy levels, ionisation energy, and electron orbit sizes. 
• Bohr’s model includes the classical description of the electron as a particle exposed to the Coulomb force and whose motion must obey Newton’s equations of motion, along with two non-classical quantisation postulates. As an isolated system, the hydrogen atom must obey the rules of conservation of energy and momentum as we know them from classical physics. 

Conclusion

The Bohr Model of the hydrogen atom aims to fill in some of the gaps identified by Rutherford’s model. Bohr proposed that electrons in an atom might orbit in stable orbits without producing radiant energy. However, the Bohr model applies only to single-electron species because the model takes only Coulombic interactions between one proton and one electron into consideration. It can’t be extended to include other atomic species with more than one electron. It can be used only with single electron species.