Discussion
The rotational spectrum of 12C16O and 13C16O provide the values of rotational constants B and B′ respectively. If B is 1.92 cm-1, what is the value of B′ ?
(h = 6.6 × 10-34 Js)
Answer: B
The rotational constant of \(^{13}C^{16}O\) is B′ = 1.83 cm\(^{-1}\).
The bond length is the same for both isotopes, so \(B = \frac{h}{8\pi^{2}c\mu r^{2}}\) is inversely proportional to the reduced mass \(\mu\):
\[ \begin{aligned} \mu &= \frac{12 \times 16}{28} = 6.857 \\ \mu' &= \frac{13 \times 16}{29} = 7.172 \\ B' &= B\frac{\mu}{\mu'} = 1.92 \times 0.956 \\ &= 1.836\ \text{cm}^{-1} \end{aligned} \]
The heavier isotope has the larger moment of inertia and so the smaller B, which rules out 2.12 cm\(^{-1}\) and 4.24 cm\(^{-1}\). 1.06 cm\(^{-1}\) is far too large a change for one extra neutron. (The value of h is not needed for a ratio.)
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