Energy Level Formula

The structure of the atom was the much-debated topic in the mid-1920s. Numerous atomic models including the theory proposed by J.J Thompson and the discovery of nucleus by Ernest Rutherford had emerged. But it was Neil Bohr, who asserted that electrons revolved around a positively charged nucleus just like the planets around the sun. In Bohr model of the hydrogen atom, an assumption concerning the quantization of atoms was made which stated that electrons orbited the nucleus in specific orbits or shells with a fixed radius. Only those shells with a radius provided by the equation below were allowed, and it was impossible for electrons to exist between these shells. Mathematically, the allowed value of the atomic radius is given by the equation

$r(n)=n^2\times r(1)$

In the next section, let us look at the formula used to calculate the energy of the electron in the nth energy level.

Energy of Electron Formula

Bohr calculated the energy of an electron in the nth level of the hydrogen atom by considering the electrons in circular, quantized orbits as

$E(n)=-\frac{1}{n^2}\times 13.6\,eV$

where 13.6 eV is the lowest possible energy of a hydrogen electron E(1).

An electron absorbs energy in the form of photons and gets excited to a higher energy level. After jumping to the higher energy level, also called the excited state, the excited electron is less stable, and therefore, would quickly emit a photon to come back to a lower and more stable energy level. The emitted energy is equivalent to the difference in energy between the two energy levels for a specific transition. The energy can be calculated using the following equation

$hv=\Delta E=(\frac{1}{n_{low}^2}-{\frac{1}{n_{high}^2}})13.6\,eV$

The formula for defining the energy levels of a hydrogen atom is given by

$E=\frac{E_0}{n^2}$

where E0 is 13.6 eV and n is 1,2,3……and so on

Summary

 Value of the Atomic Radius $r(n)=n^2\times r(1)$ Energy of an atom in the nth level of the hydrogen atom $E(n)=-\frac{1}{n^2}\times 13.6\,eV$ The value of the energy emitted for a specific transition is given by the equation $hv=\Delta E=(\frac{1}{n_{low}^2}-{\frac{1}{n_{high}^2}})13.6\,eV$ The formula for defining energy level $E=\frac{E_0}{n^2}$

Stay tuned to BYJU’S to learn more formula of various physics concepts.