Calculate the energies of the first four rotational levels of \({ }^{1} \mathrm{H}^{127} \mathrm{I}\) free to rotate in three dimensions, using for its moment of inertia \(I=\mu R^{2}\), with \(\mu=\) \(m_{\mathrm{H}} m_{1} /\left(m_{\mathrm{H}}+m_{1}\right)\) and \(R=160 \mathrm{pm}\).

When two atoms form a diatomic molecule, they create a system that can rotate about an axis. To understand the dynamics of this rotation, we need to consider the concept of reduced mass. The reduced mass, denoted as \(\mu\), simplifies the two-body problem by reducing it to an equivalent one-body problem where a single particle with mass \(\mu\) moves in the potential field of another infinitely massive particle.

The equation to calculate reduced mass is \(\mu = \frac{m_{1} m_{2}}{m_{1} + m_{2}}\), where \(m_{1}\) and \(m_{2}\) are the masses of the individual atoms within the molecule. In our exercise, where we have a hydrogen atom (\(^{1}H\)) and an iodine atom (\(^{127}I\)), the reduced mass allows us to account for the mass distribution between the two atoms during rotation. By using the atomic mass unit (amu), we standardize the mass values, facilitating easier calculations and comparisons across different molecules.

The moment of inertia is a fundamental concept in understanding rotational motion for rigid bodies. It describes how the mass of an object is distributed in relation to an axis of rotation and influences the amount of torque needed for a given angular acceleration. The equation for the moment of inertia of a diatomic molecule, where two masses rotate about a fixed point, is given by \(I = \mu R^{2}\), with \(R\) being the distance between the two atoms (bond length) and \(\mu\) being the reduced mass.

Applying this to our problem, we calculate the moment of inertia for the \(^{1}H^{127}I\) molecule using the given bond length and the previously calculated reduced mass. It's important to note that the exact value of the bond length, in this case, 160 pm, directly impacts the molecule's moment of inertia and thus its rotational characteristics.

The rigid rotor model is a simplified approach to understanding rotational motion in quantum mechanics. It assumes that the bond length between atoms in a diatomic molecule remains constant during rotation, hence the term 'rigid'. This model is instrumental in predicting the rotational energy levels of a molecule.

In the rigid rotor model, the energy of a rotational level is quantified by the equation \(E_J = \frac{\hbar^{2}}{2I}J(J+1)\), where \(E_J\) is the energy of the rotational level, \(\hbar\) is the reduced Planck's constant, \(I\) is the moment of inertia, and \(J\) is the rotational quantum number. Using this model allows students to calculate distinct energy levels and interpret spectra data for diatomic molecules, such as in our \(^{1}H^{127}I\) molecule example.

In quantum mechanics, the quantum number plays a pivotal role in determining the quantized states of a system. Specifically, for rotational motion, the rotational quantum number, \(J\), defines the state of a molecule's rotation. The values of \(J\) are non-negative integers (0, 1, 2, 3, ...), and they are directly related to the energy level of the molecule.

The significance of \(J\) lies in its dictate of energy differences between rotational levels – which are not evenly spaced due to the \(J(J+1)\) term in the energy equation. As \(J\) increases, the energy difference between consecutive levels also increases. This quantization of rotational energy underpins the production of distinct spectral lines that can be observed and measured, allowing scientists to deduce information about molecular structure and properties.