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 the 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 1.05×10 J⋅s for the value of the Reduced Planck Constant, and a value of 931.5/𝑐MeV for the value of the unified atomic mass units.

  • A7.32×10 eV
  • B7.23×10 eV
  • C7.43×10 eV
  • D7.11×10 eV
  • E7.59×10 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?

  • A2.2×10 eV
  • B3.0×10 eV
  • C2.8×10 eV
  • D2.4×10 eV
  • E3.3×10 eV

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

  • A4.9×10 eV
  • B4.6×10 eV
  • C4.5×10 eV
  • D4.2×10 eV
  • E3.8×10 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.

  • A1.47×10 eV
  • B1.61×10 eV
  • C1.26×10 eV
  • D1.40×10 eV
  • E1.14×10 eV

Determine the rotational energy of the 𝑙=2 state.

  • A4.42×10 eV
  • B3.77×10 eV
  • C4.04×10 eV
  • D3.31×10 eV
  • E2.86×10 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 N2 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 N2.

  • A2.5×10 electron volts
  • B5.7×10 electron volts
  • C1.8×10 electron volts
  • D2.0×10 electron volts
  • E1.5×10 electron volts

Q9:

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

  • A1.51×10 J
  • B4.55×10 J
  • C2.63×10 J
  • D18.2×10 J
  • E3.02×10 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.

  • A1.22×10 J
  • B7.29×10 J
  • C1.10×10 J
  • D1.47×10 J
  • E2.90×10 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.

  • A1.68×10
  • B2.98×10
  • C4.74×10
  • D4.34×10
  • E2.67×10

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?

  • A8.30×10Hz
  • B9.34×10Hz
  • C1.32×10Hz
  • D1.17×10Hz
  • E1.87×10Hz

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

  • A1.39×10J
  • B1.97×10J
  • C8.75×10J
  • D1.24×10J
  • E7.75×10J

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?

  • A6.21×10 m
  • B1.98×10 m
  • C9.89×10 m
  • D1.24×10 m
  • E3.24×10 m

Find the ground state energy of vibrations for this molecule.

  • A5.03×10 J
  • B1.60×10 J
  • C8.01×10 J
  • D5.78×10 J
  • E1.01×10 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?

  • A1.57×10 kg⋅m2
  • B3.40×10 kg⋅m2
  • C6.27×10 kg⋅m2
  • D0.965×10 kg⋅m2
  • E3.14×10 kg⋅m2

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

  • A2.90×10 J
  • B3.54×10 J
  • C2.76×10 J
  • D5.80×10 J
  • E1.38×10 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?

  • A1.6×10 kg⋅m2
  • B6.3×10 kg⋅m2
  • C1.0×10 kg⋅m2
  • D2.5×10 kg⋅m2
  • E5.1×10 kg⋅m2

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

  • A5.3×10 J
  • B8.0×10 J
  • C3.5×10 J
  • D2.7×10 J
  • E1.4×10 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.

  • A2×10 kg⋅m2
  • B6×10 kg⋅m2
  • C5×10 kg⋅m2
  • D7×10 kg⋅m2
  • E3×10 kg⋅m2

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