(I) The distance between a carbon atom \(\left( {{\bf{m = 12}}\;{\bf{u}}} \right)\) and an oxygen atom \(\left( {{\bf{m = 16}}\;{\bf{u}}} \right)\) in the CO molecule is \({\bf{1}}{\bf{.13 \times 1}}{{\bf{0}}^{{\bf{10}}}}\;{\bf{m}}\) How far from the carbon atom is the center of mass of the molecule?

The center of mass is near the heavy atom for a two-atom system.

The mass of the carbon atom is \({m_1} = 12\;{\rm{u}}\).

The mass of the oxygen atom is \({m_2} = 16\;{\rm{u}}\).

The distance between the atoms is \(r = 1.13 \times {10^{ - 10}}\;{\rm{m}}\).

Let the carbon atom be at the origin, and the oxygen atom at \(r\).

Now, the center of mass of the CO atom is:

\(\begin{array}{c}{x_{{\rm{CM}}}} = \frac{{\left( {{m_1} \times 0} \right) + \left( {{m_2} \times r} \right)}}{{{m_1} + {m_2}}}\\ = \frac{{\left( {12\;{\rm{u}} \times 0} \right) + \left( {16\;{\rm{u}} \times 1.13 \times {{10}^{ - 10}}\;{\rm{m}}} \right)}}{{12\;{\rm{u}} + 16\;{\rm{u}}}}\\ = 6.46 \times {10^{ - 11}}\;{\rm{m}}\end{array}\)

Hence, the center of mass of the CO atom is at a distance of \(6.46 \times {10^{ - 11}}\;{\rm{m}}\) from the carbon atom.