A diatomic molecule is made of two masses m1 and m2 which...

A diatomic molecule is made of two masses $m_{1}$ and $m_{2}$ which are separated by a distance $r$. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by: ( $n$ is integer) (A) $\frac{\left(m_{1}+m_{2}\right)^{2} n^{2} h^{2}}{2 m_{1}^{2} m_{2}^{2} r^{2}}$ (B) $\frac{n^{2} h^{2}}{2\left(m_{1}+m_{2}\right) r^{2}}$ (C) $\frac{2 n^{2} h^{2}}{\left(m_{1}+m_{2}\right) r^{2}}$ (D) $\frac{\left(m_{1}+m_{2}\right) n^{2} h^{2}}{2 m_{1} m_{2} r^{2}}$

A diatomic molecule is made of two masses $m_{1}$ and $m_{2}$ which are separated by a distance $r$. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by: ( $n$ is integer)
(A) $\frac{\left(m_{1}+m_{2}\right)^{2} n^{2} h^{2}}{2 m_{1}^{2} m_{2}^{2} r^{2}}$
(B) $\frac{n^{2} h^{2}}{2\left(m_{1}+m_{2}\right) r^{2}}$
(C) $\frac{2 n^{2} h^{2}}{\left(m_{1}+m_{2}\right) r^{2}}$
(D) $\frac{\left(m_{1}+m_{2}\right) n^{2} h^{2}}{2 m_{1} m_{2} r^{2}}$

Key Concepts

Rotational Energy Levels

The quantization of angular momentum directly leads to quantized rotational energy levels. The rotational energy of a diatomic molecule is derived by combining the quantized angular momentum with the classical formula for rotational kinetic energy (which involves the moment of inertia), resulting in energy levels that are proportional to the square of the quantum number.

Reduced Mass

The concept of reduced mass is used when dealing with two-body problems to simplify the analysis. For a diatomic molecule, using the reduced mass allows one to effectively treat the system as if a single mass is rotating at a certain distance, which simplifies the expressions for the moment of inertia and energy levels.

Angular Momentum Quantization

This concept states that the angular momentum of a system in quantum mechanics can only take on discrete values, typically given by multiples of a fundamental constant. In the case of a rotating diatomic molecule, Bohr’s rule is applied such that the angular momentum is quantized, leading to only specific allowed energy levels.

Moment of Inertia

In rotational dynamics, the moment of inertia quantifies how the mass is distributed relative to the axis of rotation. For a diatomic molecule, the moment of inertia is determined by the masses of the atoms and the distance between them, and it plays a crucial role in relating the angular momentum to the rotational kinetic energy.

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