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?

    • One 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
    • Two spins join the three motions
    • Pressure doubles the count

  6. Equal heat enters a rigid box and a pushing piston. Which gas ends hotter?

    • The rigid box, since all heat stays inside
    • 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
    • Spins froze out, leaving three motions
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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. One 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. Two spins join the three motions · Three drifts plus two spins make five ways, hence five halves.
  6. The rigid box, since all heat stays inside · No boundary work happens in the rigid box, so all heat lifts temperature.
  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. Spins froze out, leaving three motions · Cooling locks the spins, so only three drifts share energy.
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