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 $

Also Read :

∗ Rutherford experiment & Observations
∗ Distance of Closest Approach
∗ Bohr’s Atomic Theory
∗ Bohr’s Stationary Radii & Orbital Speed
∗ Origin of Spectra

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