Explain main postulates of Bohr's theory of atomic model. How did this theory justify the origin of spectral lines in hydrogen atom ? [4+1]

Main Postulates of Bohr’s Theory of Atomic Model

Bohr’s atomic model (1913) was proposed to explain the stability of atoms and the origin of spectral lines. Its key postulates are:

1. Stationary Orbits (Quantized Energy Levels)

   • Electrons revolve around the nucleus in fixed circular orbits called stationary states (or energy levels).
   • Each orbit has a definite energy ($E_n=-\frac{13.6}{n^2}$ eV for hydrogen), where $n=1,2,3,\dots$ (principal quantum number).
   • Electrons do not radiate energy while in these orbits (contrary to classical physics).

2. Energy Emission/Absorption During Transitions

   • An electron can jump from a lower energy orbit ($n_i$) to a higher one ($n_f$) by absorbing energy (e.g., light/heat).
   • Conversely, it emits energy (as a photon) when moving from a higher to a lower orbit: $$\Delta E=E_f-E_i=h\nu =\frac{hc}{\lambda }$$ where $h$ = Planck’s constant, $\nu$ = frequency, $\lambda$ = wavelength.

3. Angular Momentum Quantization

   • Only orbits with specific angular momentum are allowed: $$mvr=\frac{nh}{2\pi }\text{(where }m=\text{mass of electron, }v=\text{velocity, }r=\text{radius)}$$
   • This restricts electrons to discrete orbits, preventing them from spiraling into the nucleus.

4. Electron-Nucleus Attraction Balanced by Centrifugal Force

   • The electrostatic force of attraction ($F=\frac{e^2}{4\pi {ϵ}_0{r}^2}$) is balanced by the centrifugal force ($F=\frac{m{v}^2}{r}$), ensuring stable orbits.

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Justification of Spectral Lines in Hydrogen Atom

Bohr’s theory successfully explained the line spectrum of hydrogen (observed by Balmer, Lyman, etc.) through quantized energy transitions:

1. Discrete Energy Levels

   • Hydrogen’s electron can occupy only specific energy levels ($E_n=-13.6/n^2$ eV).
   • Transitions between these levels produce photons of specific wavelengths, not a continuous spectrum.

2. Photon Emission During Transitions

   • When an electron drops from a higher level ($n_i$) to a lower level ($n_f$), the energy difference is emitted as a photon: $$\frac{1}{\lambda }=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$$ where $R_H$ = Rydberg constant for hydrogen ($1.097\times 10^7{\text{m}}^{-1}$).

3. Series of Spectral Lines

   • Different series (e.g., Lyman series: $n_f=1$; Balmer series: $n_f=2$) arise from electrons falling to specific lower levels.
   • Example: The Balmer series (visible light) corresponds to transitions where $n_f=2$ and $n_i=3,4,5,\dots$.

4. Agreement with Experimental Data

   • Bohr’s formula matched the wavelengths of hydrogen’s spectral lines precisely, validating the theory.
   • For instance, the H-α line (656.3 nm) in the Balmer series is predicted by $n_i=3\to n_f=2$.

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Key Limitation: Bohr’s model worked only for hydrogen-like atoms (single-electron systems) and failed to explain multi-electron atoms or fine spectral details (later addressed by quantum mechanics).