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 9c

Chapter 42, Problem 9c

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)?

Verified step by step guidance

Step 1: Understand the relationship between rotational energy and angular velocity. The classical expression for rotational kinetic energy is given as:

K = \frac{1}{2} Iω 2

, where K is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.

Step 2: Use the quantum mechanical energy levels for rotation. For the rotational level

l = 1

, the energy is given by

E = \frac{h}{8} π^{2} \frac{l}{2}

, where h is Planck's constant, l is the quantum number, and I is the moment of inertia.

Step 3: Equate the classical rotational energy expression to the quantum mechanical energy for

l = 1

. This allows you to solve for the angular velocity

ω

Step 4: Once you have the angular velocity

ω

, calculate the rotational period using the relationship

T = \frac{2}{π} ω

, where T is the rotational period.

Step 5: Substitute the values for the moment of inertia I, Planck's constant h, and other constants into the equations to compute the rotational period. Ensure all units are consistent during substitution.

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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)Iω², where K is the rotational kinetic energy, 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 (I) is a measure of an object's resistance to changes in its rotation about an axis. It depends on the mass distribution relative to the axis of rotation. For a diatomic molecule like CO, the moment of inertia can be calculated based on the masses of the atoms and the distance between them, influencing the rotational energy levels.

Angular Velocity

Angular velocity (ω) is a vector quantity that represents the rate of rotation of an object around an axis. It is measured in radians per second and is essential for calculating rotational kinetic energy. In the context of molecular rotation, angular velocity helps determine how quickly a molecule rotates, which is directly related to its energy levels and rotational period.

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Related Practice

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 angular speed of the 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 $K = (\frac{1}{2}) I ω^{2}$ , for the $l equals 1$ level, what is the linear speed of each atom?

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Textbook Question

The average kinetic energy of an ideal-gas atom or molecule is $(\frac{3}{2}) k T$ , where $T$ is the Kelvin temperature (Chapter $18$ ). The rotational inertia of the H₂ molecule is $4.6 \times 10^{- 48}$ kg-m². What is the value of $T$ for which $(\frac{3}{2}) k T$ equals the energy separation between the $l equals 0$ and $l equals 1$ energy levels of H₂? What does this tell you about the number of H₂ molecules in the $l equals 1$ level at room temperature?

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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 42.242.242.2. If the energy of the rotating molecule is described by the classical expression K=(12)Iω2K=(\(\frac\)12)I\(\omega\)^2K=(21)Iω2, for the l=1 l = 1l=1 level, what is the rotational period (the time for one rotation)?

Verified video answer for a similar problem:

Key Concepts

Rotational Kinetic Energy

Moment of Inertia

Angular Velocity

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)?