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 \(m = 1.67 \times 10^{-27} \, \text{kg}\):
\[\mu = \frac{m \times m}{2m} = \frac{m}{2} = \frac{1.67 \times 10^{-27} \, \text{kg}}{2} = 0.835 \times 10^{-27} \, \text{kg}\]
\[r = 0.77 \times 10^{-10} \, \text{m} = 0.77 \, \text{\AA}\]
\[I = 0.835 \times 10^{-27} \, \text{kg} \times (0.77 \times 10^{-10} \, \text{m})^2 \approx 0.495 \times 10^{-47} \, \text{kg-m}^2\]