(a)
The moment of inertia of the hydrogen molecule about an axis through its center of mass and perpendicular to H-H bond.
(a)
Answer to Problem 13P
The moment of inertia of the hydrogen molecule about an axis through its center of mass and perpendicular to H-H bond is
Explanation of Solution
A hydrogen molecule makes a transition from ground level to
Write the formula for energy levels.
Here,
Refer equation (I) and find energy of
Here,
Refer equation (I) and find energy of
Here,
Refer equation (I) and find energy of
Here,
Write the formula for the energy difference between
Here,
Write the formula for the energy difference between
Here,
Subtract equation (III) from (II).
Re-write the above equation.
Re-write the above equation to obtain
Conclusion:
Substitute
The moment of inertia of the hydrogen molecule about an axis through its center of mass and perpendicular to H-H bond is
(b)
The vibrational frequency of the hydrogen molecule.
(b)
Answer to Problem 13P
The vibrational frequency of the hydrogen molecule is
Explanation of Solution
Refer section (a) and write the formula for the energy difference between
Here,
Re-write the above equation to get an expression for
Write the formula for
Here,
Conclusion:
Substitute
Substitute
The vibrational frequency of the hydrogen molecule is
(c)
The equilibrium separation distance for the molecule.
(c)
Answer to Problem 13P
The equilibrium separation distance for the molecule is
Explanation of Solution
Write the formula for the moment of inertia of the molecule.
Here,
Reduced mass of hydrogen molecule is half of the mass of it.
Here,
Re-write the above equation to get an expression for
Conclusion:
Substitute
The equilibrium separation distance for the molecule is
Want to see more full solutions like this?
Chapter 42 Solutions
Physics for Scientists and Engineers with Modern Physics
- Consider the HCl molecule, which consists of a hydrogen atom of mass 1 u bound to a chlorine atom of mass 35 u. The equilibrium separation between the atoms is 0.128 nm, and it requires 0.15 eV of work to increase or decrease this separation by 0.01 nm. (a) Calculate the four lowest rotational energies (in eV) that are possible, assuming the molecule rotates rigidly. (b) Find the molecules spring constant and its classical frequency of vibration. (Hint: Recall that U=12Kx2.) (c) Find the two lowest vibrational energies and the classical amplitude of oscillation corresponding to each of these energies. (d) Determine the longest wavelength radiation that the molecule can emit in a pure rotational transition and in a pure vibrational transition.arrow_forward2(6) Calculate the fundamental vibrational wavenumber (in cm-1) for HI molecule, if its angular vibrational frequency is 4.394×1014 s-1. Calculate the vibrational energy of the molecule in the ground state and the force constant. Assume the mass is the mass of a proton.arrow_forwardConsider 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_forward
- Consider the NaCl molecule, for which the rotational inertia is 1.30x 10-45 kg*m2. If infrared radiation with wavelength 30 μ m is Raman-scattered from a free NaCl molecule, what are the allowed wavelengths of the scattered radiation?arrow_forwardThe vibrational frequency n for Br2 is 323 cm-1 and the energy difference between its two lowest rotational energy levels, J = 0 and J = 1, is 0.164 cm-1. Calculate the relative populations of the v = 1 and v = 0 vibrational energylevels and the relative populations of the two lowest rotational energy levels for Br2 at 300 K. Comment on your results.arrow_forwardLet's consider the three atoms composing the molecule now have different masses and coordinate, while the axis of rotation is still z axis that is perpendicular to the xy plane. The first atom has a mass of 142.54 kg, with x coordinate at 3 m and y coordinate at 6 m. The second atom has a mass of 82.55 kg, with x coordinate at 1 m and y coordinate at 6 m. The third atom has a mass of 8 kg, with x coordinate at 5 m and y coordinate at 9 m. What is the moment of inertia in unit of kg m2 with respect to the x axis?arrow_forward
- 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_forwardA CO molecule is initially in the n = 2 vibrational level. If this molecule loses both vibrational and rotational energy and emits a photon, what are the photon wavelength and frequency if the initial angular momentum quantum number is l = 3?arrow_forwardWhat is the minimum energy required to go from one vibrational state to the next higher vibrational state for a molecule whose vibrations are modeled by a harmonic oscillator?arrow_forward
- Although an ordinary H2 molecule consists of two identical atoms, this is not the case for the molecule HD, with one atom of deuterium (Le., heavy hydrogen, 2H). Because of its small moment of inertia, the HD molecule has a relatively large value of E: 0.0057 eV. At approximately what temperature would you expect the rotational heat capacity of a gas of HD molecules to "freeze out," that is, to fall significantly below the constant value predicted by the equipartition theorem?arrow_forwardThe equilibrium separation between the two ions in the KCl molecule is 0.267 nm. (a) Assuming that the K+ and Cl- ions are point particles, compute the electric dipole moment of the molecule. (b) Compute the ratio of your result in (a) to the measured electric dipole moment of 5.41 x 10-29 C*m. This ratio is known as the fractional ionic character of the molecular bond.arrow_forwardWhen an OH molecule undergoes a transition from the n = 0 to the n = 1 vibrational level, its internal vibrational energy increases by 0.463 eV. Calculate the frequency of vibration and the force constant for the interatomic force. (The mass of an oxygen atom is 2.66 * 10-26 kg, and the mass of a hydrogen atom is 1.67 * 10-27 kg.)arrow_forward
- Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningModern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning