Read the following passage and answer the questions.
A glass discharge tube is filled with hydrogen gas at low pressure. A high voltage is applied across the tube, and an electric current passes through the gas. The tube then glows. A spectrometer is used to study the light that comes out of the tube. In Bohr’s model, the electron of a hydrogen atom can stay only in certain allowed energy states, called stationary states. The energy of the electron in the \(n\)th state is \(E_n=-13.6/n^{2}\ \mathrm{eV}\), where \(n = 1, 2, 3, \ldots\) In the tube, fast-moving electrons collide with the hydrogen atoms. These collisions raise some atoms to states of higher energy. Such an excited atom does not stay in that state for long. It soon falls back to a lower state and gives out a photon.
Some atoms in the tube are raised to the \(n = 3\) state. What is the shortest wavelength of light that these atoms can emit? (Take \(hc = 1240\ \mathrm{eV\,nm}\).)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 103 nm
An atom in the \( n=3 \) state can return to \( n=2 \) or go straight to \( n=1 \). The jump from \( n=3 \) to \( n=1 \) releases the largest energy. So its photon has the shortest wavelength.
Given:
\( E_1=-13.6\ \mathrm{eV} \)
\( \begin{aligned} E_3 &= -\frac{13.6}{3^{2}} \\ &= -1.51\ \mathrm{eV} \end{aligned} \)
\( hc=1240\ \mathrm{eV\,nm} \)
Energy of the photon:
\( \begin{aligned} E &= E_3-E_1 \\ &= -1.51-(-13.6) \\ &= 12.09\ \mathrm{eV} \end{aligned} \)
Wavelength of the photon:
\( \begin{aligned} \lambda &= \frac{hc}{E} \\ &= \frac{1240}{12.09} \\ &\approx 103\ \mathrm{nm} \end{aligned} \)
From the above, \( \lambda \approx 103\ \mathrm{nm} \). Therefore, the shortest wavelength these atoms can emit is about 103 nm.
NCERT (2026-27):
The Line Spectra of the Hydrogen Atom – “According to the third postulate of Bohr’s model, when an atom makes a transition from the higher energy state with quantum number ni to the lower energy state with quantum number nf (nf < ni), the difference of energy is carried away by a photon of frequency νif such that”
\( h\nu_{if}=E_{n_i}-E_{n_f} \qquad (12.11) \)
Energy levels – “… Similarly, E3 = –1.51 eV and E3 – E1 = 12.09 eV, …”