The Bohr Radius
Personally, I think the Bohr model is underrated; and yes, there are more accurate models out there to represent atoms but still the Bohr model can give us a good description of ionic, metallic, and covalent bonds as well as the spectral lines of hydrogen.
First, we need to understand that hydrogen is a simple two body system: an electron orbiting a proton in a circular orbit. The electron behaves both as a particle and a wave as represented by the de Broglie wave-equation:
Solving simultaneously for p (electron momentum) gives us the equation:
The potential (V) of the particle can be given by
The kinetic energy (K) of a paricle of known momentum and mass can be worked out with the equation K=(p^2)/2m. The total energy of the particle is given by E=K+V. This means that the energy of the electron can be given by:
...which can be represented with the arbitrary constants A and B.
The Bohr Model explains that any change in electron energy comes with a change in the orbital radius. Therefore, the energy will be of a stationary value with respect to the radius in order to form a stable energy level.
Simply solve for r, substitute A and B, and simplify:
This gives us an orbital radius of around:
Therefore, to three significant figures, the radius of a hydrogen atom is 0.531 angstroms.
First, we need to understand that hydrogen is a simple two body system: an electron orbiting a proton in a circular orbit. The electron behaves both as a particle and a wave as represented by the de Broglie wave-equation:
Solving simultaneously for p (electron momentum) gives us the equation:
The potential (V) of the particle can be given by
The kinetic energy (K) of a paricle of known momentum and mass can be worked out with the equation K=(p^2)/2m. The total energy of the particle is given by E=K+V. This means that the energy of the electron can be given by:
...which can be represented with the arbitrary constants A and B.
The Bohr Model explains that any change in electron energy comes with a change in the orbital radius. Therefore, the energy will be of a stationary value with respect to the radius in order to form a stable energy level.
Simply solve for r, substitute A and B, and simplify:
This gives us an orbital radius of around:
QED
No comments:
Post a Comment