THE LINE SPECTRA OF THE HYDROGEN ATOM -

According to the third postulate of Bohr’s model, when an atom makes a transition from the higher energy state with quantum number $\fn_cm \large n_i$ to the lower energy state with quantum number $\fn_cm \large n_f$ $\fn_cm \large (n_f<n_i)$ , the difference of energy is carried away by a photon of frequency $\fn_cm \large \nu$ , such that

$\fn_cm \large h\nu=E_f-E_i$

$\fn_cm \large \Rightarrow h\nu=\frac{-21.76\times10^{-19}Z^2}{(n_f)^2}-\frac{-21.76\times10^{-19}Z^2}{(n_i)^2}$

$\fn_cm \large \Rightarrow h\nu=21.76\times 10^{-19}Z^2\left ( \frac{1}{(n_i)^2}- \frac{1}{(n_f)^2} \right )$

$\fn_cm \large \Rightarrow \frac{hc}{\lambda}=21.76\times 10^{-19}Z^2\left ( \frac{1}{(n_i)^2}- \frac{1}{(n_f)^2} \right )$

$\fn_cm \large \therefore \frac{1}{\lambda}=\frac{21.76\times 10^{-19}Z^2}{hc} \left ( \frac{1}{(n_i)^2}- \frac{1}{(n_f)^2} \right )$

where $\fn_cm \large \frac{1}{\lambda}=\bar{\nu}$ is known as wave number (number of complete wave in one wavelength)

also $\fn_cm \large R=\frac{21.76\times 10^{-19}}{hc}$ is Rydberg constant , and if we put the value of h and c we get

$\fn_cm \large R=1.03\times10^7 m^{-1}$

$\fn_cm \large i.e \;\;\frac{1}{\lambda}=RZ^2 \left ( \frac{1}{(n_i)^2}- \frac{1}{(n_f)^2} \right )$

The above value is very close to the $\fn_cm \large 1.097\times10^7m^{-1}$ obtained from the empirical Balmer formula.” This agreement between the theoretical and experimental values of the Rydberg constant provided a direct and striking confirmation of the Bohr’s model.”

for H atom Z=1 and

$\fn_cm \large \frac{1}{\lambda}=R \left ( \frac{1}{(n_i)^2}- \frac{1}{(n_f)^2} \right )$

These spectral lines are called emission lines.

NOTE:

The explanation of the H-atom spectrum provided by Bohr’s model was a brilliant achievement, which greatly stimulated progress towards the modern quantum theory. In 1922 Bohr’s was awarded Nobel prize in physics.

IMPORTANT LINKS OF WAVE OPTICS
Introduction	Alpha Particle Scattering and Rutherford’s Nuclear Model of Atom
Atomic Spectra	Bohr Model of the Hydrogen Atom
The Line Spectra of the Hydrogen Atom	De Broglie’s Explanation of Bohr’s Second Postulate of Quantisation