All the atoms in a sample of hydrogen gas are in the \(n=4\) state. The atoms return to the ground state, directly or through lower states. How many of the following photon energies can be emitted in this process?
0.66 eV, 0.85 eV, 1.89 eV, 2.55 eV, 3.40 eV, 10.2 eV, 12.75 eV, 13.6 eV
(Take \(E_n=-13.6/n^2\ \mathrm{eV}\))
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Five
A photon carries away the energy difference between the two levels of a downward jump. From \( E_n=-13.6/n^2 \), the levels are \( E_1=-13.6 \), \( E_2=-3.40 \), \( E_3=-1.51 \) and \( E_4=-0.85 \) (all in eV).
The photon energies in the list that can be emitted:
0.66 eV — jump from \( n=4 \) to \( n=3 \)
1.89 eV — jump from \( n=3 \) to \( n=2 \)
2.55 eV — jump from \( n=4 \) to \( n=2 \)
10.2 eV — jump from \( n=2 \) to \( n=1 \)
12.75 eV — jump from \( n=4 \) to \( n=1 \)
The energies that cannot be emitted:
0.85 eV — the energy needed to free the electron from \( n=4 \); no two levels differ by this amount
3.40 eV — the energy needed to free the electron from \( n=2 \); no two levels differ by this amount
13.6 eV — the largest jump available from \( n=4 \) gives only 12.75 eV
Number of photon energies that can be emitted = 5
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”
\( h\nu_{if}=E_{n_i}-E_{n_f} \)\( \qquad (12.11) \)
Bohr Model of the Hydrogen Atom – “Atomic energies are often expressed in electron volts (eV) rather than joules. … can be rewritten as”
\( \displaystyle E_n=-\frac{13.6\ \text{eV}}{n^2} \)\( \qquad (12.10) \)