The GATE Grind

GATE 2024 PH – Question 15

Atomic and Molecular Physics · rotational and vibrational spectra of diatomic molecules · 1 mark · Multiple choice

The mean distance between atoms in HD is $r$, with hydrogen mass $m_H$. The difference between the two lowest rotational energies in multiples of $\hbar^2/(m_Hr^2)$ is

  1. $3/2$
  2. $2/3$
  3. 6
  4. $4/3$

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Correct answer: (A) $3/2$

Explanation

**Reduced mass of HD.** The masses are $m_H$ and $m_D=2m_H$ (deuterium):
$$\mu=\frac{m_Hm_D}{m_H+m_D}=\frac{2m_H^2}{3m_H}=\frac23m_H .$$

**Moment of inertia:** $I=\mu r^2=\dfrac{2}{3}m_Hr^2$.

**Rotational levels of a rigid rotor:** $E_J=\dfrac{\hbar^2}{2I}J(J+1)$.

The two lowest levels are $J=0$ ($E=0$) and $J=1$ ($E=\hbar^2/I$):
$$\Delta E=\frac{\hbar^2}{I}=\frac{\hbar^2}{\tfrac23m_Hr^2}=\frac32\cdot\frac{\hbar^2}{m_Hr^2}.$$

The difference is **3/2** in the given units (option A).