QB_PH_SEC6_016: Molecular Spectroscopy

The rotational constant of $^{12}\text{C}^{16}\text{O}$ is $B = 1.93\text{ cm}^{-1}$. Upon isotopic substitution to $^{13}\text{C}^{16}\text{O}$, what is the new rotational constant $B'$ (in $\text{cm}^{-1}$)? Take atomic masses: $^{12}\text{C} = 12.000\text{ u}$, $^{13}\text{C} = 13.003\text{ u}$, $^{16}\text{O} = 15.995\text{ u}$. (Round off your answer to two decimal places)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Rotational Constant Formula**: $B = \frac{h}{8\pi^2 c I} = \frac{h}{8\pi^2 c \mu R_e^2}$ 2. **Isotopic Invariance of Bond Length**: Under Born-Oppenheimer approximation, the electronic potential curve and equilibrium bond distance $R_e$ are unaffected by isotopic substitution. $\frac{B'}{B} = \frac{\mu}{\mu'}$ 3. **Reduced Mass Computations**: $\mu(^{12}\text{C}^{16}\text{O}) = \frac{12.000 \times 15.995}{12.000 + 15.995} = \frac{191.94}{27.995} = 6.8562\text{ u}$ $\mu'(^{13}\text{C}^{16}\text{O}) = \frac{13.003 \times 15.995}{13.003 + 15.995} = \frac{207.983}{28.998} = 7.1723\text{ u}$ 4. **New Rotational Constant**: $B' = 1.93\text{ cm}^{-1} \times \frac{6.8562}{7.1723} = 1.93 \times 0.95593 = 1.8449\text{ cm}^{-1} \approx 1.84\text{ cm}^{-1}$ 5. **Official Range**: $[1.83, 1.86]$.