Question

The inter-molecular distance between two atoms of a hydrogen molecule is $0.77 \, \text{\AA}$, and the mass of a proton is $1.67 \times 10^{-27} \, \text{Kg}$. The moment of inertia of a molecule is:

• A. $4.95 \times 10^{-47} \, \text{Kg·m}^2$
• B. $0.495 \times 10^{-47} \, \text{Kg·m}^2$
• C. $4.5 \times 10^{-47} \, \text{Kg·m}^2$
• D. $45.9 \times 10^{-47} \, \text{Kg·m}^2$

Hint:

The moment of inertia for molecular systems is crucial in determining their rotational spectra, which is foundational in molecular physics and chemistry for identifying molecular structures.

Answer 1

The Correct Option is B

To calculate the moment of inertia $I$ for a diatomic molecule such as hydrogen ($H_2$), use the formula:
$$I = \mu r^2$$
where $\mu$ is the reduced mass given by $\mu = \frac{m_1 \times m_2}{m_1 + m_2}$ for two identical masses $m$, and $r$ is the inter-molecular distance. For two protons, each with mass $m = 1.67 \times 10^{-27} \, \text{kg}$:
$$\mu = \frac{m \times m}{2m} = \frac{m}{2} = \frac{1.67 \times 10^{-27} \, \text{kg}}{2} = 0.835 \times 10^{-27} \, \text{kg}$$
$$r = 0.77 \times 10^{-10} \, \text{m} = 0.77 \, \text{\AA}$$
$$I = 0.835 \times 10^{-27} \, \text{kg} \times (0.77 \times 10^{-10} \, \text{m})^2 \approx 0.495 \times 10^{-47} \, \text{kg-m}^2$$