Name: de Broglie Hypothesis
Uploaded: 2019-07-19
Duration: 11 min 28 s
Description: The De Broglie hypothesis proposes that all matter exhibits wave-like properties and relates the observed wavelength of matter to its momentum. Let's learn.

Question

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.

$\bar{ν} = 109677 (\frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}})$

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 description of each step and each term.

Accepted Answer

The following points of Bohr's model of an atom can be used to arrive at the given formula:

(i) Electrons revolve around the nucleus in circular orbits with fixed values of energy.

(ii) When electron jumps from one orbit to another, energy is emitted or absorbed. Derivation of the given formula:

The energy of electron in the $n^{th}$ stationary state is given by

Where, $E_{n} = \frac{- 2 π^{2} {me}^{4}}{n^{2} h^{2}}$

$m =$ mass of electron

$e =$ charge on electron

$h =$ Planck's constant

When electron jumps from outer orbit $n_{f}$ to inner orbit $n_{i}$ , the difference of energy $(∆ E)$ is emitted.

$ΔE = E_{f} - E_{i} = \frac{- 2 π^{2} {me}^{4}}{n_{f}^{2} h^{2}} - (\frac{- 2 π^{2} {me}^{4}}{n_{i}^{2} h^{2}})$

$= \frac{- 2 π^{2} {me}^{4}}{h^{2}} (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$

$\bar{ν} = \frac{ΔE}{hc} = \frac{2 π^{2} {me}^{4}}{{ch}^{3}} (\frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}})$

By putting values of $π, m, c, h$ and $e$ we get,

$\bar{ν} = 109677 (\frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}})$ .

Important Questions on Structure of Atom

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