GATE 2026 PH – Question 62
A diatomic molecule has rotational lines spaced by 20.0 cm−1 and moment of inertia $I_0\times10^{-47}$ kg m². The value of $I_0$ (one decimal place) is _____. Given $h=6.6\times10^{-34}$ J s, $c=3\times10^8$ m/s.
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Correct answer: 2.79 to 2.81
Explanation
For a rigid rotor the rotational lines in the spectrum are separated by $2B$, where $B$ is the rotational constant (in wavenumber units).
**Step 1: rotational constant.**
$$2B=20.0\text{ cm}^{-1}\;\Rightarrow\;B=10.0\text{ cm}^{-1}=1000\text{ m}^{-1}.$$
**Step 2: moment of inertia** from $B=\dfrac{h}{8\pi^2cI}$:
$$I=\frac{h}{8\pi^2cB}=\frac{6.6\times10^{-34}}{8\pi^2\times3\times10^8\times1000}=\frac{6.6\times10^{-34}}{2.3687\times10^{13}}=2.786\times10^{-47}\text{ kg m}^2.$$
In units of $10^{-47}$ kg m², $I_0=\mathbf{2.8}$.