Q.What are the postulates of Bohr's model of hydrogen atom? Explain the formation of lines in the Hydrogen spectrum.

Bohr's model quantizes the electron's orbit and angular momentum; the discrete hydrogen spectral lines arise from photon emission when an electron jumps between these fixed energy levels.

Postulates of Bohr's Model of the Hydrogen Atom:

1. An electron in an atom revolves around the nucleus in certain fixed circular paths of definite energy, called stationary states or orbits. As long as the electron stays in a given orbit, it does not radiate (lose) energy — contrary to what classical electromagnetic theory would predict.

2. Only those orbits are permitted for which the angular momentum of the electron is an integral multiple of $h/2\pi$:

   $mvr = \frac{nh}{2\pi}, \quad n = 1, 2, 3, \ldots$

   where $m$ = mass of electron, $v$ = velocity, $r$ = radius of orbit, $h$ = Planck's constant, and $n$ is the principal quantum number. This is the quantization condition.

3. The energy of an electron changes only when it moves from one allowed orbit to another. When energy is supplied (e.g., by heat or an electric discharge), the electron absorbs energy and jumps to a higher-energy orbit (excited state). This excited state is unstable, and the electron falls back to a lower-energy orbit, emitting the energy difference as a photon of light. The energy of the photon equals the difference in energy between the two orbits:

   $\Delta E = E_2 - E_1 = h\nu$

4. The energy of the electron in the $n^{\text{th}}$ orbit of a hydrogen-like species is given by $E_n = -\dfrac{13.6\,Z^2}{n^2}\ \text{eV}$ (for H, $Z=1$), and the radius of the $n^{\text{th}}$ orbit is $r_n = 0.529\,n^2\ \text{Å}$.

Formation of the hydrogen spectrum:

When a hydrogen gas discharge tube is excited (e.g., by an electric spark), electrons in H atoms absorb energy and jump from the ground state ($n=1$) to various higher energy levels. Since these excited states are unstable, the electrons quickly fall back down to lower energy levels, releasing the excess energy as photons of specific frequencies/wavelengths (since the energy levels are quantized/discrete, only specific energy differences — and hence only specific wavelengths — are possible). This is why the hydrogen spectrum is a line spectrum (discrete bright lines) rather than a continuous spectrum.

Depending on which orbit the electron finally lands in, the emitted lines are grouped into named series: …