White light is passed through a gas. The spectrum of the transmitted light shows a dark line at \(620\ \mathrm{nm}\). What is the energy gap between the two levels of the gas atoms that produce this line? (Take \(h=6.6\times10^{-34}\ \mathrm{J\,s}\), \(c=3\times10^{8}\ \mathrm{m\,s^{-1}}\) and \(1\ \mathrm{eV}=1.6\times10^{-19}\ \mathrm{J}\).)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(2.0\ \mathrm{eV}\)
A dark line appears because atoms of the gas absorb photons of that wavelength. An atom absorbs a photon only if the photon energy equals the gap between a lower level and a higher level. So the gap is the energy of a \( 620\ \mathrm{nm} \) photon.
Given:
\( \lambda=620\ \mathrm{nm}=6.2\times10^{-7}\ \mathrm{m} \)
\( \begin{aligned} E &= \frac{hc}{\lambda} \\ &= \frac{6.6\times10^{-34}\times3\times10^{8}}{6.2\times10^{-7}} \\ &= 3.2\times10^{-19}\ \mathrm{J} \end{aligned} \)
In electron volts:
\( \begin{aligned} E &= \frac{3.2\times10^{-19}}{1.6\times10^{-19}} \\ &= 2.0\ \mathrm{eV} \end{aligned} \)
From the above, a photon of \( 620\ \mathrm{nm} \) carries \( 2.0\ \mathrm{eV} \). Hence, the two levels of the gas atoms are \( 2.0\ \mathrm{eV} \) apart.
NCERT (2026-27):
The Line Spectra of the Hydrogen Atom – “… But when an atom absorbs a photon that has precisely the same energy needed by the electron in a lower energy state to make transitions to a higher energy state, the process is called absorption. … The dark lines indicate the frequencies that have been absorbed by the atoms of the gas.”