Hydrogen atoms are raised to one particular excited level. As they fall back to lower levels, six different spectral lines are seen in all. What is the energy of the electron in that excited level? (Take \(E_n=-13.6/n^2\ \mathrm{eV}\).)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(-0.85\ \mathrm{eV}\)
From a level \( n \), the electron can fall to any lower level, either in one jump or in steps. Every pair of levels gives one line. So the number of different lines is the number of pairs that can be made from \( n \) levels.
\( \begin{aligned} N &= \frac{n(n-1)}{2} \\ 6 &= \frac{n(n-1)}{2} \\ n(n-1) &= 12 \\ n &= 4 \end{aligned} \)
The six lines come from the jumps 4→3, 4→2, 4→1, 3→2, 3→1 and 2→1.
Energy of the \( n=4 \) level:
\( \begin{aligned} E_4 &= -\frac{13.6}{4^2} \\ &= -0.85\ \mathrm{eV} \end{aligned} \)
From the above, six lines need four levels. Hence, the atoms were raised to \( n=4 \), where the energy is \( -0.85\ \mathrm{eV} \).
NCERT (2026-27):
The Line Spectra of the Hydrogen Atom – “The various lines in the atomic spectra are produced when electrons jump from higher energy state to a lower energy state and photons are emitted.”
Bohr Model of the Hydrogen Atom – “Atomic energies are often expressed in electron volts (eV) rather than joules. Since 1 eV = 1.6 × 10–19 J, … can be rewritten as”
\( \displaystyle E_n=-\frac{13.6\ \mathrm{eV}}{n^{2}} \)\( \qquad (12.10) \)