Energy of the orbital electron in various orbits of hydrogen atom
When the electron is in the innermost orbit of the hydrogen atom, it is said to be in its ground state with lowest energy, when the electron is in any allowed higher orbits it is said to be in its excited state with more energy.
Total energy of the electron in a privileged orbit denoted by n is En.
En = - e^2/ (8 πεorn)
Substitute the value of rn from En = - [e^2/(8 πεo)] x [mπe^2/n^2 εoh^2] = - [e^4 x m]/8 εo^2h^2] x (1/n^2)
When the electron is in the first stable orbit the system (n=1) is said to be in the ground state. The energy of the system becomes E1 = - [e^4 x m]/8 εo^2h^2]
= - [(1.602 x 10^-19)^4 x 9.108 x 10^-31]/[8 x (8.85 x 10^-12)^2 x(6.626x 10^-34 )^2] = - (1.602)^4 x 9.108 x 10^-15 / 8 x (8.85)^2 x (6.626)^2 = - 2.18 x 10^-18 j =- 2.18 x 10^-18/1.602 x 10^-19 = -13.6 e.V
When the electron is in the higher orbits with n ≥ 2, the system is said to be in the excited state. The system has less binding energy. The energy of the electron in the excited states En = - [e^4 x m] / 8 εo^2h^2] x (1/n^2) where n = 2,3,4..... or En = - 13,6 / n^2 eV
Electronic transition energy
When the electron is in any one of the excited states, automatically without any external influence, it turns into its ground state, where the potential energy is minimum . During this transition, the difference in total energy is emitted out as em waves.
ΔE = E excited /initial - E ground / final
= [ e^4 x m] / 8 εo^2h^2] [(1/nf^2 - 1/ni^2)]
The frequency of the emitted radiation ν = ΔE/h = [m e^4 /8 εo^2 h^3] [(1/nf^2 - 1/ni^2)]
The wavelength of the emitted radiation λ = C/ ν= 8 c εo^2 h^3 / (1/nf^2 - 1/ni^2) m e^4
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