When an electric discharge is passed through hydrogen gas,the hydrogen molecules dissociate to produce excited hydrogen atoms. These excited atoms emit electromagnetic radiation of discrete frequencies which can be given by the general formula
$\bar{v} = 109677 \left[ \frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}} \right] \ cm^{-1}$
What points of Bohr's model of an atom can be used to arrive at this formula? Based on these points,derive the above formula,giving a description of each step and each term.

(A) The two important points of Bohr's model that can be used to derive the given formula are as follows:
$(i)$ Electrons revolve around the nucleus in a circular path of fixed radius and energy. These paths are called orbits,stationary states,or allowed energy states.
$(ii)$ Energy is emitted or absorbed when an electron moves from a higher stationary state to a lower stationary state or from a lower stationary state to a higher stationary state,respectively.
Derivation:
The energy of an electron in the $n^{\text{th}}$ stationary state is given by: $E_{n} = -R_{H} \left( \frac{1}{n^{2}} \right)$,where $R_{H}$ is the Rydberg constant $(2.18 \times 10^{-18} \ J)$.
The energy change $(\Delta E)$ when an electron transitions from an initial orbit $(n_{i})$ to a final orbit $(n_{f})$ is:
$\Delta E = E_{f} - E_{i} = -R_{H} \left( \frac{1}{n_{f}^{2}} \right) - \left( -R_{H} \frac{1}{n_{i}^{2}} \right) = R_{H} \left[ \frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}} \right]$
Since $\Delta E = h\nu$,the frequency $\nu$ is:
$\nu = \frac{\Delta E}{h} = \frac{R_{H}}{h} \left[ \frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}} \right]$
The wavenumber $\bar{\nu}$ is defined as $\bar{\nu} = \frac{\nu}{c} = \frac{\Delta E}{hc}$. Substituting the values:
$\bar{\nu} = \frac{R_{H}}{hc} \left[ \frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}} \right]$
Using $R_{H} = 2.18 \times 10^{-18} \ J$,$h = 6.626 \times 10^{-34} \ J \ s$,and $c = 3 \times 10^{10} \ cm \ s^{-1}$,the constant $\frac{R_{H}}{hc} \approx 109677 \ cm^{-1}$,which is the Rydberg constant for wavenumber $(R)$.
Thus,$\bar{\nu} = 109677 \left[ \frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}} \right] \ cm^{-1}$.