Two atoms of cesium (Cs) can form a Cs_2 molecule. The equilibrium distance between the nuclei in a Cs_2 molecule is 0.447 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 2.21 * 106-25 kg.

Verified step by step guidance

1. First, understand that the moment of inertia (I) for a diatomic molecule like Cs_2 can be calculated using the formula: I = μ * r^2, where μ is the reduced mass of the two atoms and r is the distance between the nuclei.
2. Calculate the reduced mass (μ) of the two cesium atoms. The reduced mass is given by the formula: μ = (m1 * m2) / (m1 + m2), where m1 and m2 are the masses of the two cesium atoms. Since both atoms are cesium, m1 = m2 = 2.21 * 10^-25 kg.
3. Substitute the values into the reduced mass formula: μ = (2.21 * 10^-25 kg * 2.21 * 10^-25 kg) / (2.21 * 10^-25 kg + 2.21 * 10^-25 kg).
4. Next, convert the equilibrium distance from nanometers to meters for consistency in units. Since 1 nm = 10^-9 m, the distance r = 0.447 nm = 0.447 * 10^-9 m.
5. Finally, substitute the values of μ and r into the moment of inertia formula: I = μ * r^2, and perform the multiplication to find the moment of inertia about the specified axis.

Key Concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational motion about an axis. It depends on the mass distribution relative to the axis of rotation. For a system of particles, it is calculated as the sum of the products of each particle's mass and the square of its distance from the axis.

Center of Mass

The center of mass is the point in a system of particles where the total mass of the system can be considered to be concentrated. For two identical particles, it is located at the midpoint of the line joining them. This concept is crucial for calculating the moment of inertia in a two-particle system like Cs_2.

Equilibrium Distance

Equilibrium distance refers to the stable separation between two atoms in a molecule where the potential energy is minimized. In the context of Cs_2, this distance is 0.447 nm, and it is essential for determining the position of the atoms relative to the center of mass, which is needed to calculate the moment of inertia.