Spectral Lines

Emission Spectrum of Hydrogen

The emission spectrum of hydrogen

From experiment, the wavelengths of light emitted from a hydrogen atom are: 410nm, 434nm, 486nm, and 656nm.

The only equtions we need to know for this are E=hf (Planck's formula), and c=λf. From this, we can say that the energy of a photon (a particle of light) can be given by E=hc/λ. We can use this to calculate the energies that the photons emitted by the hydrogen atom are. Then, we will develop the mathematics behind the Bohr model and test our theory with this experimental data.

From the previous proof of the Bohr radius,  we said that the energy of an electron orbiting a nucleus in a hydrogen atom was given by the equation: E = Ar^-2 – Br^-1, and we found that r was given by the equation: r=2A/B. (A and B were arbitrary constants that we will substitute in later.)

If we substitute r=2A/B into the equation E = Ar^-2 – Br^-1, and substitute in the values for constants A and B we get the following formula.

n is the energy shell that the electron occupies and all of the other numbers are constants independant of thre state of the electron. So the formula can be expressed as:

We can assign another constant, x, to make our calculations slightly easier:

 So, now we can express an electron in an energy level n by the formula: E= -x/n², so the change in energycan be represented by the following equation:

 In order to conserve energy, the energy lossed by the electron will also be the same as the energy of the light released. So the energies of light that we got from the experimental data at the start of the proof needs to fit the formula we just made. 

 If the electron goes from the sixth energy level to the second energy level, we can find out the chnage in energy by substituting 6 for n1 and 2 for n2. Similarly, we can do this for any energy jump in the hydrogen atom. The visible photon energies from the experimental data correspond to the following electron jumps: n=6 to n=2, n=5 to n=2, n=4 to n=2, and n=5 to n=2.

Therefore, the visible emission spectrum of a hydrogen atom is due to the electron in the hydrogen atom moving between specific energy levels.

QED