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Ch 42: Molecules and Condensed Matter
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 42: Molecules and Condensed MatterProblema 9b
Capitolo 42, Problema 9b

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?

Guida verificata passo dopo passo
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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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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.
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Intro to Rotational Kinetic Energy

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.
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Intro to Moment of Inertia

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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Linear Thermal Expansion
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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 rotational period (the time for one rotation)?

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Domanda del libro di testo

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 angular speed of the rotating molecule?

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