QB_PH_SEC6_004: Molecular Spectroscopy

For a rigid diatomic molecule with rotational constant $B$, the separation between consecutive lines in its pure rotational absorption spectrum is

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Rotational Energy Term**: $F(J) = B J(J+1)$ in $\text{cm}^{-1}$. 2. **Allowed Transitions**: Under electric dipole selection rules, $\Delta J = \pm 1$. In absorption: $\bar{\nu}_{J \to J+1} = F(J+1) - F(J) = B(J+1)(J+2) - BJ(J+1) = 2B(J+1)$ 3. **Spectral Line Positions**: - For $J = 0 \to 1$: $\bar{\nu}_0 = 2B$ - For $J = 1 \to 2$: $\bar{\nu}_1 = 4B$ - For $J = 2 \to 3$: $\bar{\nu}_2 = 6B$ 4. **Constant Separation**: $\Delta \bar{\nu} = \bar{\nu}_{J+1} - \bar{\nu}_J = 2B(J+2) - 2B(J+1) = 2B$. Option (B) is correct.