Question

An oxygen molecule consists of two oxygen atoms whose total mass is \({\bf{5}}{\bf{.3 \times 1}}{{\bf{0}}^{{\bf{ - 26}}}}\;{\bf{kg}}\) and the moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is \({\bf{1}}{\bf{.9 \times 1}}{{\bf{0}}^{{\bf{ - 46}}}}\;{\bf{kg}} \cdot {{\bf{m}}^{\bf{2}}}\). From these data, estimate the effective distance between the atoms.

Short answer

The effective distance between two atoms is \(1.2 \times {10^{ - 10}}\;{\rm{m}}\).

Step-by-step solution

Given data

The moment of inertia is equal to the product of the mass and the square of the distance of the mass from the rotational axis.Here, first, find the mass of one atom and assume the effective distance between two atoms, and then find the total moment of inertial of the two-atom system.

The total mass of the oxygen atoms is \(m = 5.3 \times {10^{ - 26}}\;{\rm{kg}}\).

The moment of inertia of the atoms is \(I = 1.9 \times {10^{ - 46}}\;{\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}}\).

Let r be the effective distance between the two atoms.

Calculation

The mass of one atom is \(\frac{m}{2}\), and the distance of the atom from the rotational axis is \(\frac{r}{2}\).

The rotation axis is midway between the atoms.

Therefore, the moment of inertia of the two atoms is

\(\begin{align}I &= 2 \times \left\{ {\left( {\frac{m}{2}} \right) \times {{\left( {\frac{r}{2}} \right)}^2}} \right\}\\I &= \frac{1}{4}m{r^2}\\1.9 \times {10^{ - 46}}\;{\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}} &= \frac{1}{4} \times \left( {5.3 \times {{10}^{ - 26}}\;{\rm{kg}}} \right) \times {r^2}\\r &= 1.2 \times {10^{ - 10}}\;{\rm{m}}\end{align}\).

Hence, the effective distance between the two atoms is \(1.2 \times {10^{ - 10}}\;{\rm{m}}\).