The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition. A diatomic molecule has moment of inertia \(I\). By Bohr's quantization condition its rotational energy in the \(n^{\text {th }}\) level \((n=0\) is not allowed is A) \(\frac{1}{n^{2}}\left(\frac{h^{2}}{8 \pi^{2} I}\right)\) B) \(\frac{1}{n}\left(\frac{h^{2}}{8 \pi^{2} I}\right)\) C) \(n\left(\frac{h^{2}}{8 \pi^{2} I}\right)\) D) \(n^{2}\left(\frac{h^{2}}{8 \pi^{2} I}\right)\)

One of the fundamental tenets of quantum mechanics is the quantization of angular momentum, which was first proposed by Niels Bohr in his model of the hydrogen atom. Bohr postulated that the angular momentum of an electron orbiting around a nucleus can take on only specific discrete values, rather than any arbitrary value.

This quantization is mathematically expressed as: \[ L = n\left(\frac{h}{2\pi}\right) \] where \(L\) represents the angular momentum, \(h\) is Planck's constant, \(n\) is the principal quantum number which can take on positive integral values (1, 2, 3, ...), and \(2\pi\) is a factor from the nature of the electron's circular path.

In simple terms, quantized angular momentum means that the angular momentum of an electron is restricted to certain levels, similar to rungs on a ladder. An electron can jump from one rung to another but cannot exist between these rungs. These jumps result in the absorption or emission of energy, leading to the characteristic spectra of atoms, which was a crucial discovery for understanding atomic structure and the energy levels within an atom.

Extending Bohr's principle to diatomic molecules, we can examine the rotational energy levels that such a molecule can possess. In classical mechanics, a rotating object has continuous energy values, but quantum mechanics dictates that these energies are also quantized.

Thus, a diatomic molecule's rotational energy levels are determined by the molecule's moment of inertia (\(I\)) and its rotational quantum number (\(n\)). For a rigidly rotating diatomic molecule, Bohr's quantization condition is applied to the angular momentum, which then can be used to derive its rotational energy.

The equation of the rotational energy (\(E\)) in terms of the quantized angular momentum is expressed as: \[ E = \frac{L^2}{2I} \] where \(L\) again represents quantized angular momentum and is given by \(L = n(h/2\pi)\), and \(I\) is the moment of inertia of the molecule. As a molecule transitions between these energy levels, it can emit or absorb energy in discrete packets, known as quanta, which can be detected as spectral lines in the molecule's spectrum.

As seen in our step-by-step solution, the rotational energy of a diatomic molecule is therefore proportional to the square of the principal quantum number. The correct expression, reflecting the quantized nature of the rotational energy, is option D: \[ E = n^2\left(\frac{h^2}{8\pi^2 I}\right) \] which shows that the rotational energy is scaled by the square of the quantum number, a direct result of the quantization of angular momentum.

The principal quantum number, represented by \(n\), is a vital concept in both Bohr's model of the atom and quantum mechanics. It describes the energy level or shell in which an electron resides within an atom and takes on positive integer values (1, 2, 3, ...).

In Bohr's model, the quantization of angular momentum is directly related to the principal quantum number because it is this number that determines which allowed angular momentum states an electron can occupy. The larger the value of \(n\), the higher the energy level and the greater the distance from the nucleus at which the electron orbits.

It is also essential to note that the principal quantum number indicates the relative size and energy of atomic and molecular orbitals. The energy associated with each level is inversely proportional to the square of the principal quantum number, thus as \(n\) increases, the energy level becomes less negative, signifying that the electron is less tightly bound to the nucleus and requires less energy to escape, which is indicative of ionization.