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
- Discuss the differences between the rotational and vibrational energy levels of the deuterium (“heavy hydrogen”) molecule D2 and those of the ordinary hydrogen molecule H2. A deuterium atom has twice the mass of an ordinary hydrogen atom.arrow_forwardThe CO molecule makes a transition from the J = 1 to the J = 2 rotational state when it absorbs a photon of frequency 2.30 x 1011 Hz. (a) Find the moment of inertia of this molecule from these data.arrow_forwardConsider the hydrogen molecule H₂ as a rigid rotor with distance of separation of H-atoms r = 1.0 Å. Compute the energy of J = 2 rotational level.arrow_forward
- The effective spring constant associated with bonding in the N2 molecule is 2 297 N/m. The nitrogen atoms each have a mass of 2.32 x 10-26 kg, and their nuclei are 0.120 nm apart. Assume the molecule is rigid. The first excited vibrational state of the molecule is above the vibrational ground state by an energy difference ΔE. Calculate the J value of the rotational state that is above the rotational ground state by the same energy difference ΔE.arrow_forwardThe characteristic rotational energy for a diatomic molecule consisting of two idential atoms of mass 14 u (unified mass units) is 3.68 e-4 eV. Calculate the separation distance between the two atoms. Subarrow_forwardThis problem deals with the splitting of rotational energy levels of diatomic molecules. If one atom of the molecule has more than one stable isotope, then both isotopes are normally present in a sample. Show that the fractional change ∆f/f in the observed frequency of a photon emitted in a transition between adjacent rotational states is equal to the fractional difference in the reduced mass ∆μ/μ for molecules containing the two different isotopes.arrow_forward
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- Consider a CO molecule that is initially in the ground state of n = 0, l = 0. If the energy of a vibrational transition from the n = 0 state to the n = 1 state in CO could instead be absorbed in a rotational transition, what would be the value of l for the final state?arrow_forwardThe rotation of a molecule can be represented by the motion of a particle moving over the surface of a sphere. Based on this, calculate the magnitude of its angular momentum when l = 1 and the possible components of the angular momentum along the z-axis. Express your results as multiples of ћ. Show full and complete procedure in a clear way. DO NOT SKIP ANY STEParrow_forwardAssume the distance between the protons in the H2 molecule is 0.750 x 10-10 m. (a) Find the energy of the first excited rotational state, with J = 1. (b) Find the wavelength of radiation emitted in the transition from J = 1 to J = 0.arrow_forward
- 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