Find the wavelength of a photon emitted in the \(l=5\) to \(l=4\) transition of a molecule whose rotational inertia is \(1.75 \times 10^{-47} \mathrm{kg} \cdot \mathrm{m}^{2}\)

In the realm of quantum mechanics, the energy of a rotating molecule is quantized, meaning it can only take on specific, discrete values. These are termed as the quantum rotational energy levels. Unlike classical objects, quantum systems like molecules can't spin with just any energy; their rotational energy must correspond to certain allowed levels. Each level corresponds to a rotational quantum number, denoted as 'l' or 'J' in different contexts.

The energy for a given quantum rotational level can be expressed using the formula: \( E_{l} = hcBl(l+1) \), where \(h\) is Planck's constant, \(c\) is the speed of light, \(B\) is the rotational constant, and \(l\) is the rotational quantum number. As \(l\) changes, the molecule can transition between these levels, either absorbing or emitting a photon. In this particular exercise, the transition from \(l=5\) to \(l=4\) results in the emission of a photon, as the molecule loses energy and moves to a lower energy state.

Understanding these energy levels is crucial for numerous applications like spectroscopy, where energy transitions provide insights into molecular structures.

When a molecule transitions between two quantum rotational energy levels, it emits or absorbs a photon whose energy matches the difference between those levels. According to the principles of quantum mechanics, the energy of the emitted photon is directly related to its wavelength. To calculate the wavelength \(\lambda\) of the emitted photon when a molecule transitions between energy levels, we use the relationship \(E = \frac{hc}{\lambda}\), where \(E\) is the energy difference, \(h\) is Planck's constant, and \(c\) is the speed of light.

To find \(\lambda\), we can rearrange the equation to \(\lambda = \frac{hc}{E}\). By plugging in the value calculated for the energy difference \(\Delta E\) from the rotational energy transition, we can determine the specific wavelength of light that corresponds to that transition. This is a fundamental aspect of molecular spectroscopy and plays a pivotal role in the identification of substances and determination of molecular structure.

Molecular rotational inertia, also called the moment of inertia, is a key factor in determining how a molecule rotates. It's analogous to the mass in linear motion, representing the resistance to change in rotational state. In quantum mechanics, it directly influences the energy levels a molecule can possess when rotating. The moment of inertia \(I\) is dependent on the mass distribution within the molecule and the distances those masses are from the axis of rotation.

The formula for the rotational constant \(B\), which appears in the energy level and wavelength calculations, is derived from the moment of inertia: \(B = \frac{h}{8\pi^2cI}\). Here, \(h\) is Planck's constant and \(c\) is the speed of light. This constant serves as a bridge between the physical structure of the molecule and its energy characteristics. A molecule with a larger moment of inertia has lower rotational energy levels, which can lead to longer wavelengths of photons emitted during rotational transitions.