Bohr model

In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by electrons in orbit - similar in structure to the solar system. Because of its simplicity, the Bohr model is still commonly taught to introduce students to quantum mechanics.

Derivation of the electron energy levels of hydrogen

The Bohr model is actually only accurate for one-electron systems such as the hydrogen atom or singly-ionized helium. Here we use it to derive the energy levels of hydrogen.

Related Topics:
Hydrogen atom - Helium

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We begin with three simple assumptions:

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:1) All particles are wavelike, and an electron's wavelength lambda, is related to its velocity v by:

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:::lambda = rac{h}{m_e v}

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::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 us to make the following intuitive statement.

Related Topics:
Planck's Constant - De Broglie hypothesis

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:2) The circumference of the electron's orbit must be an integer multiple of its wavelength:

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:::2 pi r = n lambda ,

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::where r is the radius of the electron's orbit, and n is a positive integer.

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:3) The electron is held in orbit by the coulomb force. That is, the coulomb force is equal to the centripetal force:

Related Topics:
Coulomb force - Centripetal force

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:::rac{ke^2}{r^2} = rac{m_e v^2}{r} ,

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::where k = 1 / {4 pi epsilon _0}, and e is the charge of the electron.

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

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:and because of the virial theorem, the total energy simplifies to

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::E = -egin{matrix} rac{1}{2} end{matrix}m_e v^2

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Finally, we find an equation that gives us the energy of the different levels of Hydrogen:

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Thus, the lowest energy level of hydrogen (n = 1) is -13.6 eV. The next energy level (n = 2) is -3.4 eV. The third (n = 3) is -1.51 eV, and so on.

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Note that these energies are less than zero, this means that the electron is in a bound state with the proton.

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