White light is passed through hydrogen gas. All the atoms of the gas are in the ground state. Dark absorption lines are seen in the spectrum of the transmitted light. What is the longest wavelength among these dark lines? (Take \(E_n=-13.6/n^2\ \mathrm{eV}\) and \(hc=1240\ \mathrm{eV\,nm}\).)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(122\ \mathrm{nm}\)
An atom absorbs a photon only when the photon energy equals the gap between its present level and a higher level. Here every atom starts from \( n=1 \). The longest wavelength means the smallest photon energy. So it belongs to the smallest upward jump, \( n=1 \) to \( n=2 \).
Given:
\( E_1=-13.6\ \mathrm{eV} \)
\( \begin{aligned} E_2 &= -\frac{13.6}{2^2} \\ &= -3.4\ \mathrm{eV} \end{aligned} \)
Energy absorbed:
\( \begin{aligned} E &= E_2-E_1 \\ &= -3.4-(-13.6) \\ &= 10.2\ \mathrm{eV} \end{aligned} \)
Longest wavelength:
\( \begin{aligned} \lambda &= \frac{hc}{E} \\ &= \frac{1240}{10.2} \\ &= 122\ \mathrm{nm} \end{aligned} \)
From the above, the jump from \( n=1 \) to \( n=2 \) needs the least energy, \( 10.2\ \mathrm{eV} \). Hence, the longest dark line is at about \( 122\ \mathrm{nm} \).
NCERT (2026-27):
Energy levels – “… It means that the energy required to excite an electron in hydrogen atom to its first excited state, is an energy equal to E2 – E1 = –3.40 eV – (–13.6) eV = 10.2 eV.”
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.”