Ch 42: Molecules and Condensed Matter

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 42: Molecules and Condensed Matter

Problem 7b

Chapter 41, Problem 7b

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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Step 1: Recall the classical expression for rotational kinetic energy, which is given as $ K = \frac{1}{2} I \omega^2 $, where $ I $ is the moment of inertia and $ \omega $ is the angular velocity. For the quantum rotational energy levels, the energy is quantized as $ E_l = \frac{l(l+1)\hbar^2}{2I} $, where $ l $ is the rotational quantum number.

Step 2: For the $ l = 1 $ level, substitute $ l = 1 $ into the quantized energy expression $ E_l = \frac{l(l+1)\hbar^2}{2I} $ to find the rotational energy. This energy corresponds to the classical rotational kinetic energy $ K $. Equate $ K = E_l $ to find $ \omega $.

Step 3: Solve for the angular velocity $ \omega $ using the relationship $ K = \frac{1}{2} I \omega^2 $. Rearrange the equation to get $ \omega = \sqrt{\frac{2K}{I}} $. Substitute $ K = E_l $ from Step 2 into this expression.

Step 4: The linear speed $ v $ of each atom in the molecule is related to the angular velocity $ \omega $ by the formula $ v = r \omega $, where $ r $ is the distance of the atom from the axis of rotation. For a diatomic molecule like CO, $ r $ is the bond length divided by 2 for each atom.

Step 5: Substitute the values of $ \omega $ (from Step 3) and $ r $ (from the bond length of CO) into the formula $ v = r \omega $ to calculate the linear speed of each atom. Ensure that the units are consistent throughout the calculation.

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Key Concepts

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

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by an object due to its rotation. It is given by the formula K = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. This concept is crucial for understanding how energy is distributed in rotating systems, such as molecules.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution relative to the axis of rotation. For a molecule like CO, the moment of inertia can be calculated based on the masses of the atoms and their distances from the center of mass, which is essential for determining rotational energy levels.

Linear Speed

Linear speed refers to the speed at which an object moves along a path. For rotating objects, the linear speed of a point on the object can be calculated using the relationship v = rω, where r is the radius (distance from the axis of rotation) and ω is the angular velocity. Understanding linear speed is important for analyzing the motion of atoms in a rotating molecule.

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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 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

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.

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Textbook Question

The maximum wavelength of light that a certain silicon photocell can detect is $1.11$ mm. What is the energy gap (in electron volts) between the valence and conduction bands for this photocell?

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Textbook Question

Potassium bromide (KBr) has a density of $2.75 times 10 cubed$ kg/m³ and the same crystal structure as NaCl. The mass of a potassium atom is $6.49 \times 10^{- 26}$ kg, and the mass of a bromine atom is $1.33 \times 10^{- 25}$ kg. Calculate the average spacing between adjacent atoms in a KBr crystal.

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The rotational energy levels of CO are calculated in Example 42.242.2. If the energy of the rotating molecule is described by the classical expression K=(12)Iω2K=(\(\frac\)12)I\(\omega\)^2, for the l=1 l = 1 level, what is the linear speed of each atom?

Verified video answer for a similar problem:

Key Concepts

Rotational Kinetic Energy

Moment of Inertia

Linear Speed

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?