Consider the HCl molecule, which consists of a hydrogen
(a)
The four lowest rotational energies that are possible for the
Answer to Problem 9P
The four lowest rotational energies that are possible for the
Explanation of Solution
Write the expression for the rotational energy of diatomic molecule
Write the expression for the moment of inertia of diatomic molecule about its center of mass
Write the expression for the reduced mass of the molecule, Equation 11.3
Here,
Substitute equation (III) in (II)
Substitute
From the above equation, the reduced mass is
Substitute
For
For
For
For
Conclusion:
The four lowest rotational energies that are possible for the
(b)
The spring constant of the molecule and its classical frequency of vibration.
Answer to Problem 9P
The spring constant of the molecule is
Explanation of Solution
The elastic potential energy of the
Write the formula for the elastic potential energy
Write the formula for the classical frequency of vibration
Here,
Rearrange equation (IV) and substitute
Conclusion:
Substitute
The spring constant of the molecule is
Substitute
Thus, the spring constant of the molecule is
(c)
The two lowest vibrational energies and the corresponding classical amplitude of oscillation.
Answer to Problem 9P
The two lowest vibrational energies are
Explanation of Solution
Write the expression for the vibrational energy
Write the expression for the total energy of simple harmonic oscillator
Here,
Since
Substitute
Substitute
Equate equation (VI) and (VII) and substitute
Substitute
Substitute
Conclusion:
Thus, the two lowest vibrational energies are
(d)
The longest wavelength radiation that the
Answer to Problem 9P
The longest wavelength radiation that the
Explanation of Solution
Write the expression for energy using Bohr’s second postulate
The longest wavelength radiation that the
In pure rotation transition, between
From part (a), for
Substitute
In pure vibrational transition, between
From part (c), for
Substitute
Conclusion:
Thus, the longest wavelength radiation that the
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Chapter 11 Solutions
Modern Physics
- Show that the moment of inertia of a diatomic molecule composed of atoms of masses mA and mB and bond length R is equal to meffR2, where meff = mAmB/(mA+mB).arrow_forwardShow that the moment of inertia of a diatomic molecule composed of atoms of masses mA and mB and bond length R is equal to meffR2, where meff = mAmB/(mA + mB).arrow_forwardTo determine the equilibrium separation of the atoms in the HCl molecule, you measure the rotational spectrum of HCl. You find that the spectrum contains these wavelengths (among others): 60.4 mm, 69.0 mm, 80.4 mm, 96.4 mm, and 120.4 mm. (a) Use your measured wavelengths to find the moment of inertia of the HCl molecule about an axis through the center of mass and perpendicular to the line joining the two nuclei. (b) The value of l changes by +-1 in rotational transitions. What value of l for the upper level of the transition gives rise to each of these wavelengths? (c) Use your result of part (a) to calculate the equilibrium separation of the atoms in the HCl molecule. The mass of a chlorine atom is 5.81 * 10-26 kg, and the mass of a hydrogen atom is 1.67 * 10-27 kg. (d) What is the longest-wavelength line in the rotational spectrum of HCl?arrow_forward
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- Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning