Question

Explain how the emission lines in the spectrum of a gas are evidence for discrete energy levels within atoms.

Accepted Answer

Discrete energy means that the energy of a system cannot take on any arbitrary value. If a container of gas at low pressure is exposed to a strong electric field, light is emitted from the gas. This emitted light can be analysed by passing it through a prism or diffraction grating. The result is a series of bands of light at different wavelengths. The set of possible wavelengths that can be emitted is called an emission spectrum. No two elements have the same wavelengths in their spectrum. The wavelengths are like fingerprints $-$ they can be used to identify the element.

According to Bohr,

In a transition from a high to lower energy state, such that the difference in energy between the two states is $E$ , the photon emitted has a wavelength given by $:$

$\frac{h c}{λ} = E$

$\Rightarrow λ = \frac{h c}{E}$

where $λ$ is wavelength of the light $($ the photon $)$ , $c$ is the speed of light and $h$ is Planck $'$ s constant, with value $6.63 \times 10^{- 34} Js$ .

Let us assume that a hydrogen atom makes a transition from the level $n = 3$ $($ whose energy is $- 1.51 eV)$ to the level $n = 2$ $($ whose energy is $- 3.40 eV)$ . The difference in energy between these two levels is $1.89 eV$ and this is the energy that will be carried by the photon emitted in this transition.

Therefore $:$

$\frac{h c}{λ} = 1.89 eV = 1.89 \times 1.6 \times 10^{- 19} = 3.024 \times 10^{- 19} J$

$λ = \frac{6.63 \times 10^{- 34} \times 3.0 \times 10^{8}}{3.024 \times 10^{- 19}}$

$λ = 6.58 \times 10^{- 7} m$

This is in excellent agreement with the wavelength of the red line in the hydrogen spectrum, in hydrogen, all transitions from higher levels to the level $n = 2$ emit photons of visible light.

So the emission spectra of elements can be understood if we accept that electrons in atoms exist in energy levels with discrete energy.

K A Tsokos Solutions for Chapter: Atomic, Nuclear and Particle Physics, Exercise 4: Exam-style questions

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