Derivation of Energy of the nth Bohr’s Orbit

 

Derivation of Energy of the n^th Bohr’s Orbit

Basic of Derivation

The total energy of an electron revolving in any orbit around the nucleus is given by,

Calculation of K.E

The K.E. of electron with mass m revolving around the nucleus with velocity v is given by the following expression;

Now the centrifugal and centripetal forces upon the revolving electron are given as:

 

Now the centrifugal and centripetal forces upon the revolving electron are given as:

At uniform circular equilibrium motion, these two opposite forces must be equal to each other i.e.

Calculation of P.E Using definition of work

Calculation of Total Energy

As we know that

Put the value of “r” from equation (iii) to equation (ii)

E is always negative. Negative sign shows that the electron is bound to the atom and energy must be spent in order to remove it from the orbit. 

All energy states are bound states as the negative sign indicates. When n = 1; this corresponds to electron at the closest possible distance from the nucleus and at its lowest energy and is called ground state energy. All energy states with value of n higher than 1 are termed as excited states. When n = α then E = 0; which means that the system is unbound and the electron is free. It should be noted that the energy is increasing as the n (orbits) increasing; however the difference of energy between two orbits is decreasing.

 

Conclusion

If total energy = - x

Then

KE = + x

PE = - 2x