QB_PH_SEC6_025: Molecular Spectroscopy

Which of the following diatomic molecules has NO electric-dipole allowed pure rotational absorption spectrum in the microwave region?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Gross Selection Rule for Rotational Absorption**: Pure rotational absorption transitions require a non-vanishing electric dipole transition matrix element: $\langle J+1 | \vec{\mu} | J \rangle \neq 0$ This condition is satisfied if and only if the molecule has a **permanent electric dipole moment** $\mu_0 \neq 0$. 2. **Symmetry Analysis of Candidate Molecules**: - $\text{CO}$: Heteronuclear $\implies \mu_0 = 0.11\text{ D} \neq 0$ (Microwave Active) - $\text{HCl}$: Heteronuclear $\implies \mu_0 = 1.08\text{ D} \neq 0$ (Microwave Active) - $\text{NO}$: Heteronuclear $\implies \mu_0 = 0.16\text{ D} \neq 0$ (Microwave Active) - $\text{N}_2$: Homonuclear centro-symmetric $\implies \mu_0 \equiv 0$ (Microwave Inactive) 3. **Conclusion**: $\text{N}_2$ has no rotational absorption spectrum. Option (A) is correct.