Energy of the electron in nth orbit

The total energy of an electron is the sum of potential and kinetic energies.

$\displaystyle P.E = -\frac{1}{4\pi\epsilon_0}\frac{(Ze)(e)}{r_n}$

$\displaystyle K.E = \frac{1}{2}mv^2$

also , $\displaystyle \frac{mv^2}{r_n} = \frac{1}{4\pi\epsilon_0}\frac{(Ze)(e)}{r_n^2}$

Total Energy En = P.E + K.E

$\displaystyle E_n = \frac{1}{4\pi\epsilon_0}\frac{(Ze)(e)}{r_n} + \frac{1}{8\pi\epsilon_0}\frac{(Ze)(e)}{r_n}$

$\displaystyle E_n = -\frac{1}{8\pi\epsilon_0}\frac{(Ze)(e)}{r_n}$

Putting the value of rn

$\displaystyle E_n = -\frac{Z^2 e^4 m}{8\epsilon_0^2 n^2 h^2}$

Energy of an electron in the ground state of hydrogen atom

Energy of the electron in the nth orbit of hydrogen atom can be given by the expression

$\displaystyle E_n = -\frac{13.6}{n^2} eV$

Energy of the electron in the nth orbit of hydrogen like atom with atomic number Z can be given by the expression.

$\displaystyle E_n = -\frac{13.6 Z^2}{n^2} eV$