Problem 10 - University Physics with Modern Physics

Question

Two atoms of cesium (Cs) can form a Cs \(_{2}\) molecule. The equilibrium distance between the nuclei in a \(\mathrm{Cs}_{2}\) molecule is 0.447 \(\mathrm{nm} .\) Calculate the moment of inertia about an axis through the center of mass of the two nuclei and perpendicular to the line joining them. The mass of a cesium atom is 2.2 \(\mathrm{I} \times 10^{-25} \mathrm{kg}\) .

Step-by-Step Solution

Answer

The moment of inertia is approximately \( 5.4925 \times 10^{-45} \text{ kg} \cdot \text{m}^2 \).

Step 1: Calculate the Reduced Mass

To find the reduced mass \( \mu \) of the Cs \(_2\) molecule, we use the formula: \( \mu = \frac{m_1 m_2}{m_1 + m_2} \). Here, both \( m_1 \) and \( m_2 \) are equal to the mass of a cesium atom, which is \( 2.2 \times 10^{-25} \) kg. Thus, \( \mu = \frac{(2.2 \times 10^{-25})(2.2 \times 10^{-25})}{2.2 \times 10^{-25} + 2.2 \times 10^{-25}} \). Simplifying this, \( \mu = \frac{(2.2 \times 10^{-25})^2}{2 \times 2.2 \times 10^{-25}} = \frac{2.2 \times 10^{-25}}{2} = 1.1 \times 10^{-25} \) kg.

Step 2: Determine the Distance to the Center of Mass

Since the atoms in the Cs \(_2\) molecule are identical, the center of mass is exactly in the middle of the bond connecting them. Therefore, the distance from the center of mass to either atom \( r \) is half of the equilibrium distance between the nuclei, which is \( \frac{0.447 \text{ nm}}{2} = 0.2235 \text{ nm} \). Convert this to meters: \( 0.2235 \text{ nm} = 0.2235 \times 10^{-9} \text{ m} = 2.235 \times 10^{-10} \text{ m} \).

Step 3: Calculate the Moment of Inertia

The moment of inertia \( I \) can be calculated using the formula \( I = \mu r^2 \), where \( \mu \) is the reduced mass and \( r \) is the distance to the center of mass. Plug in the values we have: \( I = 1.1 \times 10^{-25} \text{ kg} \times (2.235 \times 10^{-10} \text{ m})^2 \). Simplify: \( I = 1.1 \times 10^{-25} \times 4.993225 \times 10^{-20} \), giving us \( I \approx 5.4925 \times 10^{-45} \text{ kg} \cdot \text{m}^2 \).

Key Concepts

Reduced Mass

Understanding reduced mass is crucial when analyzing systems of two interacting objects, such as the Cs \(_2\) molecule. When two cesium atoms bond to form a molecule, they interact in a way that simplifies the description of their motion. Instead of dealing with two separate masses, we use a concept known as reduced mass. This is a simplified representation of their effective mass when calculating things like moment of inertia.
Think of reduced mass as a way to make the complex interactions between two identical objects more calculable. The formula for reduced mass \( \mu \) is \( \mu = \frac{m_1 m_2}{m_1 + m_2} \). Here, both \( m_1 \) and \( m_2 \) represent the mass of each cesium atom. In our exercise, this value simplifies to \( 1.1 \times 10^{-25} \text{ kg} \).

• The reduced mass helps when studying molecular vibrations and rotations.
• It's useful in calculating how energy is distributed between two interacting bodies.

By using reduced mass, problems involving two interacting masses become much easier to solve.

Equilibrium Distance

In molecular physics, the term 'equilibrium distance' refers to the perfect distance between the nuclei of two bonded atoms at which the forces of attraction and repulsion balance each other. For the Cs \(_2\) molecule, this distance is 0.447 nm. It's an important figure because it represents the most stable configuration of the molecule.
At the equilibrium distance, the potential energy of the system is minimized. This means the molecule is energetically favored to stay at this distance unless influenced by outside forces, like increased temperature or external fields. Understanding this concept allows us to predict molecular behavior under different conditions.

• The equilibrium distance plays a crucial role in determining the physical and chemical properties of a molecule.
• Knowing this distance is essential for calculating the moment of inertia, as it helps determine how the molecules will rotate around their center of mass.

The calculations of moment of inertia in our exercise rely heavily on knowing this equilibrium distance.

Cesium Molecule

The cesium molecule, represented by Cs \(_2\), is a diatomic molecule consisting of two cesium atoms. Cesium itself is a soft, gold-colored alkali metal known for its reactivity and its use in diverse applications such as atomic clocks and photoelectric cells. When two cesium atoms come together, they form a molecule that exhibits unique properties inherent to diatomic elements.
The Cs \(_2\) molecule is particularly valuable in studies of rotational and vibrational dynamics due to its simplicity. The molecule exists in a gaseous state at high temperatures and behaves much like other homonuclear diatomic molecules.

• Its symmetry results in interesting spectroscopic properties.
• Studying it helps refine our understanding of molecular interactions.

Examining the Cs \(_2\) molecule allows us to better understand basic molecular physics principles, such as reduced mass and moment of inertia.