GATE 2022 PH – Question 50

Atomic and Molecular Physics · rotational and vibrational spectra of diatomic molecules · 2 marks · Multiple select

For molecular mass M, electron mass m, and electronic, rotational, vibrational energies Ee, ER, EV, which scalings hold?

$E_R\sim(m/M)E_e$
$E_R\sim\sqrt{m/M}E_e$
$E_V\sim\sqrt{m/M}E_e$
$E_V\sim(m/M)^{1/4}E_e$

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Correct answer: (A) $E_R\sim(m/M)E_e$; (C) $E_V\sim\sqrt{m/M}E_e$

Explanation

Estimate the three kinds of energy in a molecule of size $a$ (about the Bohr radius), with $m$ the electron mass and $M$ the nuclear (molecular) mass.

**Electronic energy:** an electron confined to size $a$ has $E_e\sim\dfrac{\hbar^2}{ma^2}$.
**Rotational energy:** $E_R\sim\dfrac{\hbar^2}{I}\sim\dfrac{\hbar^2}{Ma^2}=\dfrac mM\,E_e$. ✓ (A)
**Vibrational energy:** the force constant is $k\sim E_e/a^2$, so $E_V=\hbar\sqrt{\dfrac kM}\sim\hbar\sqrt{\dfrac{E_e}{Ma^2}}=\sqrt{\dfrac mM}\,E_e$ (using $\hbar^2/a^2=mE_e$). ✓ (C)

So $E_R\sim(m/M)E_e$ and $E_V\sim\sqrt{m/M}\,E_e$. Answer **A and C**.