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Bohr model - Electron energy levels in hydrogen

Bohr model - Electron energy levels in hydrogen: Encyclopedia II - Bohr model - Electron energy levels in hydrogen

The Bohr model is accurate only for one-electron systems such as the hydrogen atom or singly-ionized helium. This section uses the Bohr model to derive the energy levels of hydrogen. The derivation starts with three simple assumptions: 1) All particles are wavelike, and an electron's wavelength λ, is related to its velocity v by: where h is Planck's Constant, and me is the mass of the ...

Bohr model: Encyclopedia II - Bohr model - Electron energy levels in hydrogen

Bohr model - Electron energy levels in hydrogen

The Bohr model is accurate only for one-electron systems such as the hydrogen atom or singly-ionized helium. This section uses the Bohr model to derive the energy levels of hydrogen.

The derivation starts with three simple assumptions:

1) All particles are wavelike, and an electron's wavelength λ, is related to its velocity v by: where h is Planck's Constant, and m_e is the mass of the electron. Bohr did not make this assumption (known as the de Broglie hypothesis) in his original derivation, because it hadn't been proposed at the time. However it allows the following intuitive statement. 2) The circumference of the electron's orbit must be an integer multiple of its wavelength: where r is the radius of the electron's orbit, and n is a positive integer. 3) The electron is held in orbit by the coulomb force. That is, the coulomb force is equal to the centripetal force: where k = 1 / 4πε₀, and q_e is the charge of the electron.

These are three equations with three unknowns: λ, r, v. After solving this system of equations to find an equation for just v, it is placed into the equation for the total energy of the electron:

Because of the virial theorem, the total energy simplifies to

Substituting, one obtains the energy of the different levels of hydrogen:

Or, after plugging in values for the constants,

Thus, the lowest energy level of hydrogen (n = 1) is about -13.6 eV. The next energy level (n = 2) is -3.4 eV. The third (n = 3) is -1.51 eV, and so on. Note that these energies are less than zero, meaning that the electron is in a bound state with the proton. Positive energy states correspond to the ionized atom where the electron is no longer bound, but is in a scattering state.

Bohr model - Energy in terms of other constants

Starting with what we found above,

We can multiply top and bottom by c², and we'll arrive at

From here we can now write the energy level equation in terms of other constants to:

where,

is the energy level is the rest energy of the electron is the fine structure constant is the principal quantum number.

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1913, 1925, 19th century, Bohr radius, Ernest Rutherford, Franck-Hertz experiment, Inert pair effect, Lyman series, Niels Bohr, Planck's Constant, Planck's constant, Rydberg formula, Schroedinger, Schrödinger equation, Stark effect, Zeeman effect, angular momentum, atom, atomic physics, atomic spectra, atoms, centripetal force, classical mechanics, coulomb force, de Broglie hypothesis, eV, electric charge, electric discharges, electromagnetic waves, electron, electrons, electrostatic forces, emission lines, energy, energy level, fine structure, fine structure constant, gasses, generalized coordinate, generalized momentum, gravity, helium, hydrogen, hydrogen atom, hyperfine structure, ionized, lithium, magnetic fields, nucleus, obsolete scientific theory, perturbation, perturbation theory, photon, planetary mechanics, principal quantum number, quantized, quantum jump, quantum jumps, quantum mechanical, quantum mechanics, rest energy, scattering state, solar system, spectroscopy, synchrotron radiation, virial theorem, wave mechanics

Adapted from the Wikipedia article "Electron energy levels in hydrogen", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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