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The existence of line spectra can be explained by means of the first “quantum” model of the atom, developed by Bohr in 1913. Although the Bohr model of the hydrogen atom was eventually replaced, it yields the correct values for the observed spectral lines and gives a substantial insight into the structure of atoms in general. The following derivation has the objective of obtaining the energy levels of the Bohr atom. If the energy levels can be obtained, then the hydrogen atom spectra can be reproduced.
The derivation proceeds with three major elements: first, use the force balance to relate the velocity to the radius of the electron orbit, then use a quantum assumption to get the radius, and then solve for the energies.
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