State the mathematical relation that is used to calculate the radius, 𝑟, of the orbit of an electron in a hydrogen atom. Give your answer in terms of the wavelength, 𝜆, associated with the electron and the electronic energy level, 𝑛.
We can start out by sketching the electron energy levels of the hydrogen atom according to the Bohr model. The 𝑛 equals one, two, and three energy levels can be sketched this way as concentric circles. We want to calculate the radii of these different circles in terms of the wavelength of the electrons in those orbits and the electron energy level 𝑛.
For each given orbital radius, we know that the circumference of that circular path is equal to two times 𝜋 times 𝑟. That’s the distance along which the electrons waveform exists. We see from our drawing of the hydrogen atom that the various energy levels of this atom are quantized. They occur in discrete quantities.
Something very similar happens — we may recall — with waves on a string. There are certain allowable resonant frequencies of that wavelength, given the boundary conditions. Given a total string length 𝐿 and various progressive standing waves represented by the index 𝑛, we saw that the 𝑛th wavelength of our standing wave on a string was equal to two 𝐿 divided by that index 𝑛.
Something similar happens with the allowed wavelength of electrons in the hydrogen atom. In the hydrogen atom, the wavelength of the electron in the 𝑛 equals one orbital is equal to two times 𝜋 times the radius 𝑟 sub one of that orbit divided by that index 𝑛. Likewise, the wavelength of the electron in the 𝑛 equals two orbit is equal to two times 𝜋 times that radial distance divided by two the index 𝑛.
This trend continues as we move to higher and higher energy levels. So multiplying both sides of this equation by the electronic energy level 𝑛, we can see that two times 𝜋 times the radius of the electron orbital is equal to the electronic energy level times the allowed wavelength. This is the mathematical relation describing the quantization of electron orbital radius.