Worksheet: Molecular Energy Levels

In this worksheet, we will practice calculating molecular rotational and vibrational energies, atomic equilibrium separations, and dissociation energies.

Q1:

The characteristic energy of the N2 molecule is 2.48×10 eV. Determine the separation distance between the nitrogen atoms.

Q2:

Find the equilibrium separation distance between the Na+ and F ions in an NaF crystal. Use a value of 41.99 g/mol for the molar mass of NaF and use a value of 2.56 g/cm3 for the density of an NaF crystal.

Q3:

The separation between hydrogen atoms in a H2 molecule is about 0.075 nm. Determine the characteristic energy of rotation in eV. Use a value of 1.05×10Js for the value of the Reduced Planck Constant, and a value of 931.5 MeV/c2 for the value of the unified atomic mass unit.

  • A 7 . 2 3 × 1 0 eV
  • B 7 . 3 2 × 1 0 eV
  • C 7 . 4 3 × 1 0 eV
  • D 7 . 5 9 × 1 0 eV
  • E 7 . 1 1 × 1 0 eV

Q4:

The crystal structure of caesium iodide is body-centered cubic. A Cs+ ion occupies a cubic volume of 𝑟, where 𝑟 is the equilibrium separation of ions in the crystal. What is the distance of a Cs+ ion to its “nearest neighbor” I ion if 𝑟=0.46 nm?

Q5:

A diatomic F molecule with an equilibrium separation of 0.14 nm is in the 𝑙=1 state.

What is the energy of the molecule?

  • A 2 . 2 × 1 0 eV
  • B 3 . 0 × 1 0 eV
  • C 2 . 8 × 1 0 eV
  • D 2 . 4 × 1 0 eV
  • E 3 . 3 × 1 0 eV

How much energy is radiated in a transition from a 𝑙=2 to a 𝑙=1 state?

  • A 4 . 9 × 1 0 eV
  • B 4 . 6 × 1 0 eV
  • C 4 . 5 × 1 0 eV
  • D 4 . 2 × 1 0 eV
  • E 3 . 8 × 1 0 eV

Q6:

An H2 molecule with an equilibrium separation distance of 0.0750 nm can have various rotational energy states.

Determine the rotational energy of the 𝑙=0 state. Given the mass of Hydrogen 1.674×10 kg

Determine the rotational energy of the 𝑙=1 state.

  • A 1 . 4 7 × 1 0 eV
  • B 1 . 6 1 × 1 0 eV
  • C 1 . 2 6 × 1 0 eV
  • D 1 . 4 0 × 1 0 eV
  • E 1 . 1 4 × 1 0 eV

Determine the rotational energy of the 𝑙=2 state.

  • A 4 . 4 2 × 1 0 eV
  • B 3 . 7 7 × 1 0 eV
  • C 4 . 0 4 × 1 0 eV
  • D 3 . 3 1 × 1 0 eV
  • E 2 . 8 6 × 1 0 eV

Q7:

A molecule oscillates at a frequency of 88 THz. What is the difference between its adjacent energy levels?

Q8:

The separation between nitrogen atoms in an N molecule is 0.11 nm. Determine the characteristic energy of rotation in electron volts. Use a value of 14 u for the atomic mass of N.

  • A 1 . 8 × 1 0 eV
  • B 2 . 5 × 1 0 eV
  • C 2 . 0 × 1 0 eV
  • D 5 . 7 × 1 0 eV
  • E 1 . 5 × 1 0 eV

Q9:

The potential energy of a crystal is 9.10 eV per ion pair. Find the dissociation energy for three moles of the crystal.

  • A 1 . 5 1 × 1 0 J
  • B 4 . 5 5 × 1 0 J
  • C 2 . 6 3 × 1 0 J
  • D 1 8 . 2 × 1 0 J
  • E 3 . 0 2 × 1 0 J

Q10:

Molecular hydrogen is kept at a temperature of 300 K in a cubical container with sides each 15.0 cm long. Consider the molecules as though they are moving in a one-dimensional box. Take the mass of one mole of hydrogen to be 2.01588 g.

Find the ground state energy of a hydrogen molecule in the container.

  • A 1 . 2 2 × 1 0 J
  • B 7 . 2 9 × 1 0 J
  • C 1 . 1 0 × 1 0 J
  • D 1 . 4 7 × 1 0 J
  • E 2 . 9 0 × 1 0 J

Assume that a molecule has a thermal energy given by 𝑘𝑇2 and find the corresponding quantum number 𝑛 of the quantum state to this thermal energy.

  • A 1 . 6 8 × 1 0
  • B 2 . 9 8 × 1 0
  • C 4 . 7 4 × 1 0
  • D 4 . 3 4 × 1 0
  • E 2 . 6 7 × 1 0

Q11:

Vibrations of an H2 molecule can be modeled as a simple harmonic oscillator with the spring constant 𝑘=1.15×10/Nm, and the mass of the hydrogen atom is 𝑚=1.67×10kg. The molecule makes a transition between its third and second excited states.

What is the vibrational frequency of this molecule before it makes the transition?

  • A 8 . 3 0 × 1 0 H z
  • B 9 . 3 4 × 1 0 H z
  • C 1 . 3 2 × 1 0 H z
  • D 1 . 1 7 × 1 0 H z
  • E 1 . 8 7 × 1 0 H z

What is the energy of the photon emitted during the transition?

  • A 1 . 3 9 × 1 0 J
  • B 1 . 9 7 × 1 0 J
  • C 8 . 7 5 × 1 0 J
  • D 1 . 2 4 × 1 0 J
  • E 7 . 7 5 × 1 0 J

What is the wavelength of the photon emitted during the transition?

  • A14.3 μm
  • B2.27 μm
  • C1.60 μm
  • D0.256 μm
  • E10.0 μm

Q12:

A diatomic molecule behaves like a quantum harmonic oscillator with the force constant 15.0 N/m and mass 6.50×10 kg. The molecule makes the transition from the third excited state to the second excited state, during which it emits a photon.

What is the wavelength of the photon?

  • A 6 . 2 1 × 1 0 m
  • B 1 . 9 8 × 1 0 m
  • C 9 . 8 9 × 1 0 m
  • D 1 . 2 4 × 1 0 m
  • E 3 . 2 4 × 1 0 m

Find the ground state energy of vibrations for this molecule.

  • A 5 . 0 3 × 1 0 J
  • B 1 . 6 0 × 1 0 J
  • C 8 . 0 1 × 1 0 J
  • D 5 . 7 8 × 1 0 J
  • E 1 . 0 1 × 1 0 J

Q13:

In a physics lab, you measure the vibrational–rotational spectrum of potassium bromide (KBr). The estimated separation between the lowest absorption peaks Δ𝑓=5.35×10 Hz, and the central frequency of the band 𝑓=9.70×10 Hz.

What is the moment of inertia of a KBr molecule?

  • A 1 . 5 7 × 1 0 kg⋅m2
  • B 3 . 4 0 × 1 0 kg⋅m2
  • C 6 . 2 7 × 1 0 kg⋅m2
  • D 0 . 9 6 5 × 1 0 kg⋅m2
  • E 3 . 1 4 × 1 0 kg⋅m2

What is the energy of vibration of a KBr molecule of the lowest energy level?

  • A 2 . 9 0 × 1 0 J
  • B 3 . 5 4 × 1 0 J
  • C 2 . 7 6 × 1 0 J
  • D 5 . 8 0 × 1 0 J
  • E 1 . 3 8 × 1 0 J

Q14:

The vibrational–rotational spectrum of HCl is measured in a lab. The estimated separation between absorption peaks Δ𝑓=6.6×10Hz. The central frequency of the band 𝑓=8.0×10Hz.

What is the moment of inertia of an HCl molecule?

  • A 1 . 6 × 1 0 kg⋅m2
  • B 6 . 3 × 1 0 kg⋅m2
  • C 1 . 0 × 1 0 kg⋅m2
  • D 2 . 5 × 1 0 kg⋅m2
  • E 5 . 1 × 1 0 kg⋅m2

What is the energy of vibration of an HCl molecule of the lowest energy level?

  • A 5 . 3 × 1 0 J
  • B 8 . 0 × 1 0 J
  • C 3 . 5 × 1 0 J
  • D 2 . 7 × 1 0 J
  • E 1 . 4 × 1 0 J

Q15:

Transitions in the rotational energy spectrum of a molecule are observed at a temperature of 300 K. If a peak in the spectrum corresponds to a transition from the 𝑙=2 state to the 𝑙=1 state, what is the moment of inertia of the molecule? Answer to one significant figure.

  • A 2 × 1 0 kg⋅m2
  • B 6 × 1 0 kg⋅m2
  • C 5 × 1 0 kg⋅m2
  • D 7 × 1 0 kg⋅m2
  • E 3 × 1 0 kg⋅m2

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.