Question

Explain line spectrum of hydrogen atom on the basis of Bohr's atomic model.

Accepted Answer

Bohr explained the line spectrum of the hydrogen atom by assuming that the electron moved in circular orbits and that orbits with only certain radii were allowed. When the electron moves from one allowed orbit to another it emits or absorbs photons of energy matching exactly the separation between the energies of the given orbits (emission/absorption spectrum). We see these photons as lines of coloured light in emission or dark lines in absorption.

The energy of an electron in a particular orbit is-

$E_{n} = \frac{- R h c}{n^{2}}$

where, $R$ is the Rydberg constant, $h$ is Planck’s constant, $c$ is the speed of light, and $n$ is a principal quantum number.

According to third postulate of Bohr, when an electron jumps from higher energy level $E_{n_{2}}$ to lower energy level $E_{n_{1}}$ the emitted energy is given by equation-

$E_{n_{2}} - E_{n_{1}} = h ν = \frac{h c}{λ} . . . . . . . . (1)$

$E_{n_{2}} - E_{n_{1}} = - (\frac{R h c}{{n_{2}}^{2}}) - (- \frac{R h c}{{n_{1}}^{2}}) = - R h c (\frac{1}{{n_{2}}^{2}} - \frac{1}{{n_{1}}^{2}}) . . . . . . . (2)$

where, $n_{1}$ is the final orbit and $n_{2}$ the initial orbit.

Substituting equation $(1)$ in above equation $(2)$ , we get

$E_{n_{2}} - E_{n_{1}} = \frac{h c}{λ} = - R h c (\frac{1}{{n_{2}}^{2}} - \frac{1}{{n_{1}}^{2}})$

$\frac{1}{λ} = - R (\frac{1}{{n_{2}}^{2}} - \frac{1}{{n_{1}}^{2}})$

Bohr calculated the value of $R$ from fundamental constants such as the charge and mass of the electron and Planck's constant and obtained a value of $1.097 \times 10^{7} m^{- 1}$ , the same number Rydberg had obtained by analyzing the emission spectra. The negative sign indicates that energy is released as the electron moves from orbit $n_{2}$ to orbit $n_{1}$ because orbit $n_{2}$ is at a higher energy than orbit $n_{1}$ .

For hydrogen atom- $\frac{1}{λ} = R [\frac{1}{{n_{2}}^{2}} - \frac{1}{{n_{1}}^{2}}]$

If in this equation, $n_{2} = 2$ and $n_{1} = n$ where $n = 3, 4, 5 . . . . .$ then we get

$\frac{1}{λ} = R [\frac{1}{2^{2}} - \frac{1}{n^{2}}] n = 3, 4, 5 . . . . .$

This equation is exactly the same as Balmer and Rydberg mentioned on the basis of experience. This result is a very big achievement of Bohr's theory. In this way by putting values of $n_{1}$ and $n_{2}$ in above equation we can get expression for different series example like Lyman series, Balmer series, Paschen series, Brackett series, Pfund series.

Explain line spectrum of hydrogen atom on the basis of Bohr's atomic model.

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