The H2 molecule has a moment of inertia of 4.6×10−484.6\$$times10^{-48} kg-m2. $$What is the wavelength ll of the photon absorbed when H2 makes a transition from the l=3l = 3 to the l=4l = 4 rotational level?

Ch 42: Molecules and Condensed Matter

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 42: Molecules and Condensed Matter

Problem 4

Chapter 41, Problem 4

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?

Verified step by step guidance

Step 1: Understand the rotational energy levels of a diatomic molecule. The energy of a rotational level is given by the formula:

E

_l = ll+1 h2π I, where

l

is the rotational quantum number,

h

is Planck's constant, and

I

is the moment of inertia.

Step 2: Calculate the energy difference between the rotational levels

l = 3

and

l = 4

. Use the formula for the energy difference:

ΔE = E

₄ - E₃. Substitute the values of

l

into the energy formula for each level.

Step 3: Express the energy difference in terms of frequency using the relationship

E = hν

, where

ν

is the frequency of the photon absorbed. Rearrange the formula to solve for

ν

ν = \frac{ΔE}{h}

Step 4: Relate the frequency of the photon to its wavelength using the speed of light equation:

c = λν

, where

c

is the speed of light,

λ

is the wavelength, and

ν

is the frequency. Rearrange the formula to solve for

λ

λ = \frac{c}{ν}

Step 5: Substitute the known values for the moment of inertia

I = 4.6 × 10

^-48 kg·m², Planck's constant

h = 6.626 × 10

^-34 J·s, and the speed of light

c = 3 × 10

⁸ m/s into the equations to calculate the wavelength

λ

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Key Concepts

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

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a molecule like H2, it quantifies how mass is distributed relative to the axis of rotation. The moment of inertia affects the energy levels of rotational states, which are crucial for understanding transitions between these states.

Rotational Energy Levels

In quantum mechanics, molecules can occupy discrete rotational energy levels, denoted by quantum numbers (l). The energy associated with these levels is given by the formula E_l = (h^2 * l(l + 1)) / (2 * I), where h is Planck's constant and I is the moment of inertia. Transitions between these levels involve the absorption or emission of photons.

Photon Wavelength

The wavelength of a photon is related to its energy by the equation E = h * c / λ, where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength. When a molecule transitions between rotational levels, the energy difference corresponds to the wavelength of the absorbed or emitted photon, allowing us to calculate λ based on the energy change.

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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

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

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

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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The H2 molecule has a moment of inertia of 4.6×10−484.6\(\times\)10^{-48} kg-m2. What is the wavelength ll of the photon absorbed when H2 makes a transition from the l=3l = 3 to the l=4l = 4 rotational level?

Verified video answer for a similar problem:

Key Concepts

Moment of Inertia

Rotational Energy Levels

Photon Wavelength

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?