The Boltzmann Distribution over Molecular Levels · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

How molecules share energy levels

Science · Chemistry · ages 19-21
Name ______________________   Date ____________

  1. Why is the first excited vibrational state nearly empty at room temperature?

    • Vibrational gaps dwarf thermal energy
    • Molecules refuse to vibrate at all
    • Degeneracy is always zero for vibrations

  2. How do the populations of two levels compare?

    • The higher level always holds more
    • Degeneracy ratio times an exponential falloff with the gap over temperature
    • Both levels always hold equal shares

  3. At a given temperature, each single state holds a bigger share than any state of higher energy.

    Circle one:   True   False

  4. Two levels have equal energy, but one has twice the degeneracy. How do populations compare?

    • Equal populations
    • The doubly degenerate level holds half as many molecules
    • The doubly degenerate level holds twice the molecules

  5. Why are rotational levels heavily populated at the same temperature where vibrations sit empty?

    • Rotational gaps are small next to thermal energy
    • Rotations ignore the Boltzmann rule
    • Rotational gaps are large next to thermal energy

  6. A sample is heated strongly. What happens to the level populations?

    • The distribution sharpens and the ground state gains share
    • The distribution flattens and higher levels gain share
    • Populations freeze exactly where they were

  7. Where does the Boltzmann factor itself come from?

    • It is an arbitrary fitting constant
    • It comes from the mass of the molecules
    • Counting arrangements shows the exponential spread is the most probable one

  8. A student expects strong infrared hot bands from excited vibrations at room temperature. What is wrong?

    • Hot bands are forbidden by selection rules
    • Excited vibrational states are nearly empty, so almost nothing absorbs from them
    • Infrared light cannot reach excited molecules
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Answer key

For grown-ups. Fold this page away before handing over the rest.

How molecules share energy levels W1-mt_-DHIox6GBm-s1

  1. Vibrational gaps dwarf thermal energy · The gap is far larger than what the temperature can supply.
  2. Degeneracy ratio times an exponential falloff with the gap over temperature · The gap and the temperature set the ratio.
  3. True · Per state, the share falls exponentially as energy rises. Only degeneracy can let a higher level hold more in total.
  4. The doubly degenerate level holds twice the molecules · More states means more ways to occupy the level.
  5. Rotational gaps are small next to thermal energy · Small gaps admit many molecules while wide gaps admit almost none.
  6. The distribution flattens and higher levels gain share · Higher temperature feeds the upper levels.
  7. Counting arrangements shows the exponential spread is the most probable one · Probability picks the winner by sheer numbers.
  8. Excited vibrational states are nearly empty, so almost nothing absorbs from them · Absorption starts from the ground state because that is where the molecules are.
Worksheet · LightMySky