Question

Obtain an expression for the energy of an electron in the nth orbit of hydrogen atom in terms of the radius of the orbit and absolute constants. 

Answer 1

Consider an electron of mass m and charge –e revolving round the nucleus of an atom of atomic number Z in the nth orbit of radius ‘r’. Let ν be the velocity of the electron. The electron possess potential energy because it is in the electrostatic field of the nucleus it also possess kinetic energy by virtue of its motion. 

Potential energy of the electron is given by Ep = (potential at a distance r from the nucleus)(-e) 

Kinetic energy of the electron is given by 

E_k = 1/2 mv² --------- (2) 

From Bohr’s postulate

Substituting this value of mv² in equation (2) 

Total energy of the electron resolving in the nth orbit is given by E_n = E_p + E_k 

The radius of n^th permitted orbit of the electron is given by 

Substituting this value of r in equation (4). 

for hydrogen like atoms. 

For hydrogen atom Z = 1. 

Total energy of the electron in the nth orbit of hydrogen atom is