Derivation of Rydberg Formula for Hydrogen Atom Spectrum :
According to Bohr's model of the atom, energy is radiated in the form of a photon when an electron jumps from a higher energy state to a lower energy state. In other words, energy is radiated in the form of a photon when an electron jumps from an orbit of higher energy ($n_i$) to the lower energy orbit ($n_f$), where $n_i > n_f$. The energy of the emitted radiation or photon is given by:
$h\nu = E_{ni} - E_{nf}$
Using
$E_n = \frac{-m e^4}{8 \varepsilon_0^2 h^2 n^2}$
we get
$h\nu = \frac{-m e^4}{8 \varepsilon_0^2 h^2 n_i^2}- \left(\frac{-m e^4}{8 \varepsilon_0^2 h^2 n_f^2}\right)$
$h\nu = \frac{m e^4}{8 \varepsilon_0^2 h^2} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
$\nu = \frac{m e^4}{8 \varepsilon_0^2 h^3}\left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
Using $c = \lambda \nu$, i.e. $\nu = \frac{c}{\lambda}$ in Eq. (2), we get
$\frac{c}{\lambda} = \frac{m e^4}{8 \varepsilon_0^2 h^3}\left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
$\frac{1}{\lambda} = \frac{m e^4}{8 \varepsilon_0^2 c h^3}\left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
$\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
where $\lambda$ is the wavelength of the spectral line emitted and
$R = \frac{m e^4}{8 \varepsilon_0^2 c h^3} = 1.0974 \times 10^7 \,\text{m}^{-1}$
is Rydberg's constant.
Equation (3) is known as the $\textbf{Rydberg Formula}$ for the hydrogen atom spectrum.
Wave Number
As $\bar{\nu} = \frac{1}{\lambda}$, i.e. wave number, we can write
$\bar{\nu} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
Using eqn. (3) we can find the wavelengths of the spectral lines emitted by hydrogen atom.
SHORT CUT: Rydberg formula can be used in a convenient form as
$\lambda = \frac{911 \text{Å}}{\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)}$
NUMERICAL EXAMPLE 5:
The wavelength of yellow line of sodium is 5896 Å. What is its wave number?
Solution:
Wave number,
$\bar{\nu} = \frac{1}{\lambda}$
$\bar{\nu} = \frac{1}{5896 \times 10^{-10}}$
$\bar{\nu} = 1696100 \,\text{m}^{-1}$
$\bar{\nu} =16961 \,\text{cm}^{-1}$