Question

The distance between a carbon atom \((m=12.0 \mathrm{u} ; 1 \mathrm{u}=1\) atomic mass unit) and an oxygen atom \((m=16.0 \mathrm{u})\) in a carbon monoxide \((\mathrm{CO})\) molecule is \(1.13 \cdot 10^{-10} \mathrm{~m}\). How far from the carbon atom is the center of mass of the molecule?

Short answer

Answer: The center of mass of the carbon monoxide molecule is approximately \(6.46 \times 10^{-11} \mathrm{m}\) from the carbon atom.

Step-by-step solution

Convert atomic mass unit to kg

Before we calculate the center of mass, we need to convert the mass of the carbon and oxygen atoms from atomic mass units (u) to kg, using the conversion factor: \(1 \mathrm{u} = 1.6605 \times 10^{-27} \mathrm{kg}\) $$m_\text{C} = 12.0 \mathrm{u} \times \frac{1.6605 \times 10^{-27} \mathrm{kg}}{1 \mathrm{u}} = 1.9926 \times 10^{-26} \mathrm{kg}$$ $$m_\text{O} = 16.0 \mathrm{u} \times \frac{1.6605 \times 10^{-27} \mathrm{kg}}{1 \mathrm{u}} = 2.6568 \times 10^{-26} \mathrm{kg}$$

Calculate the center of mass of the molecule

Now we can calculate the center of mass of the carbon monoxide molecule, using the formula for the center of mass mentioned above: $$x_\text{cm} = \frac{m_\text{C} \cdot 0 + m_\text{O} \cdot (1.13 \times 10^{-10} \mathrm{m})}{m_\text{C} + m_\text{O}}$$ $$x_\text{cm} = \frac{2.6568 \times 10^{-26} \mathrm{kg} \cdot (1.13 \times 10^{-10} \mathrm{m})}{1.9926 \times 10^{-26} \mathrm{kg} + 2.6568 \times 10^{-26} \mathrm{kg}}$$

Simplify and find the center of mass

Simplify the expression and calculate the center of mass: $$x_\text{cm} = \frac{2.6568 \times 10^{-26} \mathrm{kg} \cdot (1.13 \times 10^{-10} \mathrm{m})}{4.6494 \times 10^{-26} \mathrm{kg}}$$ $$x_\text{cm} = 6.4574 \times 10^{-11} \mathrm{m}$$ So, the center of mass of the carbon monoxide molecule is approximately \(6.46 \times 10^{-11} \mathrm{m}\) from the carbon atom.

Key concepts

Atomic Mass Unit Conversion

Understanding atomic mass unit conversion is crucial when dealing with molecular physics and chemistry. An atomic mass unit (u) is a standard unit of mass that quantifies the mass of an atom or molecule. It is defined as one twelfth of the mass of a carbon-12 atom, and its value in kilograms is approximately 1.6605 x 10^-27 kg.
In the context of calculating the center of mass of a carbon monoxide (CO) molecule, it is necessary to convert the masses of carbon and oxygen from atomic mass units to kilograms. This makes it possible to apply Newtonian mechanics, which are usually expressed in SI units, to solve the problem.

• Carbon atom mass conversion:
  By multiplying 12.0 u (the atomic mass) by the conversion factor, we obtain the mass of a carbon atom in kilograms (\(m_\text{C} = 12.0 \, \text{u} \times 1.6605 \times 10^{-27} \, \text{kg/u} = 1.9926 \times 10^{-26} \, \text{kg}\)).
• Oxygen atom mass conversion:
  Similarly, converting 16.0 u (the atomic mass of oxygen) to kilograms yields (\(m_\text{O} = 16.0 \, \text{u} \times 1.6605 \times 10^{-27} \, \text{kg/u} = 2.6568 \times 10^{-26} \, \text{kg}\)).

This step is fundamental as it allows for the application of physical laws to microscopic scales using standard macroscopic units.

Molecular Physics

Molecular physics is the study of the physical properties of molecules, the forces between them, their interactions, and the laws governing their behavior. In the case of the carbon monoxide molecule, molecular physics allows us to understand its structure and calculate properties such as the center of mass.

Application to the Center of Mass

In molecular physics, the center of mass is a point that represents the average position of all the mass in a molecule. For a diatomic molecule like carbon monoxide, it is the balance point between the two atoms where the mass is equally distributed, considering their respective masses.
When calculating the center of mass in a diatomic molecule, we use the formula\[x_\text{cm} = \frac{m_\text{C} \cdot d_\text{C} + m_\text{O} \cdot d_\text{O}}{m_\text{C} + m_\text{O}}\]where \(d_\text{C}\) and \(d_\text{O}\) are the distances from the reference point (usually one of the atoms) to the center of mass for the carbon and oxygen atoms, respectively. Here, \(d_\text{C} = 0\) because the reference is chosen at the carbon atom.

Carbon Monoxide Molecule

The carbon monoxide molecule is of particular interest in both environmental science and molecular physics. It is composed of one carbon atom and one oxygen atom, bonded together by a triple bond, which places them at a specific distance from one another.

Properties and Bonding

CO is a linear molecule with a bond length of approximately 1.13 x 10^-10 m, which is the distance between the carbon and oxygen atoms. This distance is critical when calculating the center of mass of the molecule.
The molecule's center of mass lies closer to the oxygen atom due to its greater mass compared to carbon. This mass asymmetry causes the center of mass to shift toward the heavier oxygen atom. This detail has a significant impact on the behavior and reactivity of the molecule, making an understanding of its physical properties essential for students studying molecular physics and chemistry.