Ch 42: Molecules and Condensed Matter

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 42: Molecules and Condensed Matter

Problem 8

Chapter 42, Problem 8

Two atoms of cesium (Cs) can form a $C s sub 2$ molecule. The equilibrium distance between the nuclei in a $C s sub 2$ 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 $2.21 \times 10^{- 25}$ kg.

Verified step by step guidance

Step 1: Understand the problem. The moment of inertia (I) for a diatomic molecule like Cs₂ is calculated about an axis through the center of mass and perpendicular to the line joining the two nuclei. The formula for the moment of inertia is I = μ * r², where μ is the reduced mass of the system, and r is the distance between the two nuclei.

Step 2: Calculate the reduced mass (μ). 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 identical, μ simplifies to μ = m / 2, where m is the mass of a single cesium atom.

Step 3: Convert the equilibrium distance (r) into meters. The given equilibrium distance is 0.447 nm. Convert this to meters by multiplying by 10⁻⁹, so r = 0.447 * 10⁻⁹ m.

Step 4: Substitute the values of μ and r into the formula for the moment of inertia. Use the expression I = μ * r², where μ is the reduced mass calculated in Step 2 and r is the equilibrium distance in meters from Step 3.

Step 5: Simplify the expression to find the moment of inertia. Ensure that the units are consistent (kg·m²) and that the calculation is performed correctly. The result will give the moment of inertia of the Cs₂ molecule about the specified axis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specific axis. It depends on the mass distribution relative to the axis of rotation. For two point masses, the moment of inertia can be calculated using the formula I = m * r^2, where m is the mass and r is the distance from the axis of rotation.

Center of Mass

The center of mass is the point at which the mass of a system is concentrated and about which the mass is evenly distributed. For two identical masses, the center of mass lies exactly halfway between them. In the case of the Cs_2 molecule, the center of mass is crucial for calculating the moment of inertia as it defines the axis of rotation.

Equilibrium Distance

The equilibrium distance is the distance between two atoms or molecules at which the potential energy is minimized, and the forces acting on them are balanced. In the context of the Cs_2 molecule, this distance (0.447 nm) is essential for determining the separation between the two cesium nuclei when calculating the moment of inertia.

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Textbook Question

The rotational energy levels of CO are calculated in Example $42.2$ . If the energy of the rotating molecule is described by the classical expression $K=($\frac$12)I$\omega$^2$ , for the $l = 1$ level, what is the rotational period (the time for one rotation)?

The rotational energy levels of CO are calculated in Example $42.2$ . If the energy of the rotating molecule is described by the classical expression $K = (\frac{1}{2}) I ω^{2}$ , for the $l equals 1$ level, what is the linear speed of each atom?

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Textbook Question

During each of these processes, a photon of light is given up. In each process, what wavelength of light is given up, and in what part of the electromagnetic spectrum is that wavelength? A molecule decreases its vibrational energy by $0.198$ eV.

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Textbook Question

The H₂ molecule has a moment of inertia of $4.6 \times 10^{- 48}$ kg-m². What is the wavelength $l$ of the photon absorbed when H₂ makes a transition from the $l equals 3$ to the $l equals 4$ rotational level?

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Textbook Question

For the H₂ molecule the equilibrium spacing of the two protons is $0.074$ nm. The mass of a hydrogen atom is $1.67 \times 10^{- 27}$ kg. Calculate the wavelength of the photon emitted in the rotational transition $l equals 2$ to $l equals 1$ .

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