
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, 3rd Edition
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