Equipartition and the Heat Capacities of Gases · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Counting the ways a molecule stores energy

Science · Thermal & Statistical Physics · ages 18-19
Name ______________________   Date ____________

  1. What is the molar Cv of a monatomic gas?

    • R
    • 2R
    • 3R over 2

  2. What energy does one degree of freedom hold on average?

    • Half kT per molecule
    • kT times the molar mass
    • Zero at any temperature

  3. For an ideal gas, Cp is smaller than Cv.

    Circle one:   True   False

  4. A room temperature diatomic gas has Cv of 5R over 2. How many active degrees of freedom does each molecule have?

    Answer: ______________

  5. Why is room temperature diatomic Cv equal to 5R over 2?

    • Vibrations vanish entirely
    • Rotations add two ways
    • Pressure doubles the count

  6. Two sealed containers hold equal amounts of the same ideal gas at the same starting temperature. Equal heat flows into each. One container is rigid, so its gas does no expansion work. The other has a freely pushing piston and loses no other heat. Which gas ends hotter?

    • The rigid box keeps all energy for warming
    • The piston case, pushing warms extra
    • Both finish at the same temperature

  7. A student predicts every gas has Cv of 3R over 2 at all temperatures. What is the error?

    • R changes with temperature
    • Heat capacity counts active ways, which differ by molecule and temperature
    • Cv always equals Cp

  8. A cold diatomic gas measures Cv near 3R over 2. What happened?

    • The gas leaked half its atoms
    • New vibrations woke up
    • Rotations froze out
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Answer key

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

Counting the ways a molecule stores energy W1-mt_3OjJwVWlYD-s1

  1. 3R over 2 · Three drift directions give three halves of R.
  2. Half kT per molecule · Energy splits evenly, one half kT per way per molecule.
  3. False · Cp beats Cv by R because expansion work costs extra heat.
  4. 5 · Cv counts halves of R, so five halves mean five ways.
  5. Rotations add two ways · Three drifts plus two rotations make five ways, hence five halves.
  6. The rigid box keeps all energy for warming · No boundary work happens in the rigid box, so all that energy warms the gas.
  7. Heat capacity counts active ways, which differ by molecule and temperature · Structure and temperature decide how many ways are awake to share energy.
  8. Rotations froze out · Cooling freezes out the rotations, so only three drifts share energy.
Worksheet · LightMySky