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Equipartition theorem
Intermediate
Each quadratic degree of freedom has ½kT of energy.
E = ½k_BT per dofIn thermal equilibrium each quadratic term in the energy gets ½kT on average. Thus U = (f/2)NkT and C_v = (f/2)R.
Diatomic gas: C_v = 5R/2.
Fails when quantum effects freeze modes.
In more detail
The equipartition theorem states that each quadratic degree of freedom carries an average energy of ½kT. It predicts the specific heats of gases from their degrees of freedom.
Real-world uses
- Estimating heat capacities of gases in engines.
- Explaining why the specific heat of solids is roughly constant.
- Physicists use equipartition to predict the heat capacity of gases from the number of ways their molecules can store energy.
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Frequently asked questions
What is a quadratic degree of freedom?
A variable whose energy depends on its square, like kinetic energy in one direction.
Does equipartition always hold?
It fails at low temperatures where quantum effects freeze out some modes.
Does equipartition hold at very low temperatures?
No. At low temperatures, some degrees of freedom are frozen out, so the simple rule fails and quantum effects matter.
Related topics in Kinetic Theory & Statistical Physics
Mean free path · Collision frequency · Degrees of freedom · Specific heats of gases · Maxwell–Boltzmann distribution · Most probable speed