One Shot Formulae on “Structure of Atom”-02
If (Z) = atomic Number of Hydrogen like species (having one electron only) and (n) is the stationary state or orbit or energy state,
Assumptions by Bohr
Possible angular momentum of stationary state, (m_evr) = ( n. {h over 2pi}); where (n) is positive integer i.e. (n = 1,2,3,4…)
Radius
Radius of (n^{th}) orbit ((r_n)} = (left ( frac{n^2}{Z} right ) a_0), where (a_0) = 0.529 A°
Energy
Kinetic Energy (K.E.) of an electron in (n^{th}) orbit = ( 2.18 times 10^{-18} left( frac{Z^2}{n^2} right ) J) = (13.6 left( frac{Z^2}{n^2} right ) eV)
Potential Energy (P.E.) of an electron in (n^{th}) orbit = -( 4.36 times 10^{-18} left( frac{Z^2}{n^2} right ) J) = ( – 27.2 left( frac{Z^2}{n^2} right ) eV)
Magnitude of Potential Energy of an electron in (n^{th}) orbit = Magnitude of (2 times) Kinetic Energy of an electron in (n^{th}) orbit
| P.E. | = | 2 ( times) K.E. |
Total Energy of an electron in (n^{th}) orbit = Kinetic Energy of an electron in (n^{th}) orbit + Potential Energy of an electron in (n^{th}) orbit
Total Energy of an electron in (n^{th}) orbit = -( 2.18 times 10^{-18} left( frac{Z^2}{n^2} right ) J) = ( – 13.6 left( frac{Z^2}{n^2} right ) eV) = -( R_H left( frac{Z^2}{n^2} right ) ) ; where (R_H = 2.18 times 10^{-18} J)
Velocity
Velocity of an electron in (n^{th}) orbit (V_n) = (2.19 times 10^{6} left( frac{Z}{n} right ) ms^{-1})
Spectrum (Emitted/Absorbed rays)
Energy of the photon emitted/absorbed when an electron jumps from orbit (n_i) to orbit (n_f) = (Delta E) = -( 2.18 times 10^{-18} left( {1 over {n_i}^2} – {1 over {n_f}^2} right ) J); The difference of total energy.
Frequency (nu) of ray emitted when an electron jumps from orbit (n_i) to orbit (n_f) is ( 3.29 times 10^{15} left( {1 over {n_f}^2} – {1 over {n_i}^2} right ) Hz) ; where (n_i) > (n_f)
Wavenumber of emitted ray = (bar{nu}) = ( 1 over lambda ) = ( 1.09677 3.29 times 10^{7} left( {1 over {n_f}^2} – {1 over {n_i}^2} right ) m^{-1})
Frequency (nu) of ray absorbed when an electron jumps from orbit (n_i) to orbit (n_f) is ( 3.29 times 10^{15} left( {1 over {n_i}^2} – {1 over {n_f}^2} right ) Hz) ; where (n_i) < (n_f)
Wavenumber of absorbed ray = (bar{nu}) = ( 1 over lambda ) = ( 1.09677 3.29 times 10^{7} left( {1 over {n_i}^2} – {1 over {n_f}^2} right ) m^{-1})
| Series | (n_1) lower orbit | (n_2) upper orbit | Spectral Region |
|---|---|---|---|
| Lyman | 1 | 2,3,4… | Ultraviolet |
| Balmer | 2 | 3,4,5… | Visible |
| Paschen | 3 | 4,5,6… | Infrared |
| Brackett | 4 | 5,6,7… | Infrared |
| Pfund | 5 | 6,7,8… | Infrared |
