Calculate the energy change required for an electron to move between states: a quantum jump up or down an energy-level diagram.

Electrons of an atom always revolve following certain paths around the nucleus called orbits or call these Electron Shells. Each orbit has its own energy-specific energy levels. The orbit closer to the nucleus has a low energy level, and the orbit far away from the nucleus has high energy levels.

The principal Quantum number (n) can be represented as the specific orbit number belonging to a particular electron state.

There is a relationship between the energy level (E) of the electron and the closeness of the orbit to the nucleus that is given by,

$E=-\frac{13.6}{n^2}{Z}_{eff}^{2}eV$ ….. (1)

Here, is the atomic number.

Observe here that the energy level is negative with increasing n , which should always be a positive integer, i.e. $(n=1,2,3,\dots ,n)$. Hence the values of E will become as,

$\begin{array}{l}{E}_1=-13.6eV\\ E_2=-3.39eV\\ E_3=-1.51eV......\end{array}$

Hence, the energy level is quantized.

Consider a Hydrogen atom that has an atomic number (Z= 1) . The electron present in orbit with the quantum number $\left(n_1\right)$, which has the energy level $\left(E_1\right)$, gains energy to move to the next orbit with the quantum number $\left(n_2\right)$, which has the energy level $\left(E_2\right)$.

Then, the change in energy can be written from the equation (1) as:

$$\begin{gathered} ∆E=E_2-E_1 \\ =\frac{13.6}{n_2^2}{Z}_{eff}^{2}eV-\left(-\frac{13.6}{n_1^2}{Z}_{eff}^{2}eV\right) \\ =13.6\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)Z_{eff}^{2}eV \end{gathered}$$

For elements other than Hydrogen, we can write equation (1) as,

$∆E=13.6\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)Z_{eff}^{2}eV$