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.
Given below are two statements:
Statement I: In Bohr’s model, the frequency of the radiation emitted in a transition is equal to the frequency of revolution of the electron in its initial orbit.
Statement II: When an atom makes a transition between two levels with a larger energy difference, the emitted radiation has a shorter wavelength.
In the light of the above statements, choose the most appropriate answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Statement I is incorrect but Statement II is correct
Statement I In Bohr’s model, the frequency of the radiation emitted in a transition is equal to the frequency of revolution of the electron in its initial orbit. — False. This is what classical electromagnetic theory expects. In Bohr’s model, the frequency of revolution is not connected to the frequency of the spectral line. The line frequency depends on the energy difference of the two levels, \( \nu=(E_i-E_f)/h \). The two agree only for transitions between very large values of n.
Statement II When an atom makes a transition between two levels with a larger energy difference, the emitted radiation has a shorter wavelength. — True. The photon carries the energy difference, \( h\nu=E_i-E_f \), and its wavelength is
\( \begin{aligned} \lambda &= \frac{c}{\nu} \\ &= \frac{hc}{E_i-E_f} \end{aligned} \)
A larger energy difference gives a higher frequency, so the wavelength is shorter.
NCERT (2026-27):
Points to Ponder – “6. In Bohr model, contrary to ordinary classical expectation, the frequency of revolution of an electron in its orbit is not connected to the frequency of spectral line. The later is the difference between two orbital energies divided by h. For transitions between large quantum numbers (n to n – 1, n very large), however, the two coincide as expected.”
Bohr Model of the Hydrogen Atom – “(iii) Bohr’s third postulate incorporated into atomic theory the early quantum concepts that had been developed by Planck and Einstein. It states that an electron might make a transition from one of its specified non-radiating orbits to another of lower energy. When it does so, a photon is emitted having energy equal to the energy difference between the initial and final states. The frequency of the emitted photon is then given by
\( h\nu=E_i-E_f \qquad (12.6) \)
where Ei and Ef are the energies of the initial and final states and Ei > Ef.”