Sunday, February 1, 2015

Bohr’s atomic model and Radius of Orbit

As Rutherford Alfa scattering model failed to explain all the properties of the matter, a new concept called Bohr’s atomic model is introduced. This atomic model is valid for hydrogen atom and the hydrogen like atoms. Plank’s quantum concept is taken into consideration to explain this atomic model. This model has some basic postulates.

According to Bohr’s atomic model, electron can revolve around the nucleus only in specified orbits. These orbits are called stationery orbits. When the electron is revolving in this orbit, it neither loses the energy nor gains the energy. These orbits are circular in nature. 

For the electron to take that circular path it needs some centripetal force. Centripetal force is never an outside force. The force that is acting in the system towards the Centre of the system is called centripetal force. In this case the electric force of attraction between the electron and the protons of the nucleus provides the necessary centripetal force.

According to second postulate, the revolving electron has a constant angular momentum which can be expressed in terms of Planck’s constant.

According to third Bohr’s postulate, energy of the electron in the given orbit is constant. It poses both potential and kinetic energies. The sum of these total energies is always constant and the electron is not going to radiate any energy when it is orbiting in a stationary orbit.

When the electron jumps from the higher orbit to lower orbit, it releases energy. If the electron has to jump from the lower orbit to higher orbit it need energy. This energy is either emitted or absorbed in the form of wave packets called quanta. This energy will have a certain frequency and wavelength. The energy emitted or absorbed is in the integral multiples of product of Planck’s constant and the emitted frequency.

Determination of the radius of Bohr orbit

According to Bohr’s concept, the electron is revolving in a specified orbit around the nucleus. These orbits are called stationery orbits and when the electron is in this orbit it neither loose the energy nor gain the energy. For the electron to continue in the circular path it need some centripetal force. This centripetal force is provided by the electric force of attraction between the electron and the protons of the nucleus.

The electron in a circular path will always have a constant angular momentum. Taking these two things into consideration we can derive the equation for the orbital of an electron as shown below.

It can be noticed that the radius of the electron depends on the principal quantum number, mass of the electron and atomic number. It can be observed that radius is directly proportional to Squire of the principal quantum number, inversely proportional to both mass and atomic number.

For the hydrogen atom we can substitute atomic number is equal to 1. As all the terms of the equation of the radius ground state orbit are constant, we can calculate the radius of the first orbit and it can be mathematically shown that it is numerically equal to 
0.53 Å. This value is called Bohr’s radius and the corresponding orbit is called Bohr’s orbit.

Taking this value into consideration we can write the numerical value of the further orbits as shown in the above diagram.

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