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Bohr’s Atomic Model
Electron can revolve in certain nonradiating orbits called stationary or bits for which the angular momentum of electron is an integer multiple of (h / 2π)
mvr = nh / 2π
where n = I, 2. 3,… called principle quantum number.
The radiation of energy occurs only when any electron jumps from one permitted orbit to another permitted orbit.
Energy of emitted photon
hv = E_{2} – E_{1}
where E_{1} and E_{2}are energies of electron in orbits.
Radius of orbit of electron is given by
r = n^{2}h^{2} / 4π^{2} mK Ze^{2} ⇒ r ∝ n^{2} / Z
where, n = principle quantum number, h = Planck’s constant, m = mass of an electron, K = 1 / 4 π ε, Z = atomic number and e = electronic charge.
Velocity of electron in any orbit is given by
v = 2πKZe^{2} / nh ⇒ v ∝ Z / n
Frequency of electron in any orbit is given by
v = KZe^{2} / nhr = 4π^{2}Z^{2}e^{4}mK^{2} / n^{3} h^{3}
⇒ v prop; Z^{3} / n^{3}
Kinetic energy of electron in any orbit is given by
E_{k} = 2π^{2}me^{4}Z^{2}K^{2} / n^{2} h^{2} = 13.6 Z^{2} / n^{2} eV
Potential energy of electron in any orbit is given by
E_{p} = – 4π^{2}me^{4}Z^{2}K^{2} / n^{2} h^{2} = 27.2 Z^{2} / n^{2} eV
⇒ E_{p} = ∝ Z^{2} / n^{2}
Total energy of electron in any orbit is given by
E = – 2π^{2}me^{4}Z^{2}K^{2} / n^{2} h^{2} = – 13.6 Z^{2} / n^{2} eV
⇒ E_{p} = ∝ Z^{2} / n^{2}
Wavelength of radiation emitted in the radiation from orbit n_{2} to n_{1} is given by
In quantum mechanics, the energies of a system are discrete or quantized. The energy of a particle of mass m is confined to a box of length L can have discrete values of energy given by the relation
E_{n} = n^{2} h^{2} / 8mL^{2} ; n < 1, 2, 3,…
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