Equipartition theorem

Definition

The equipartition theorem states that each degree of freedom in a system at thermal equilibrium contributes an average energy of $\frac{1}{2}k_BT$ per particle, where $k_B$ is Boltzmann's constant and $T$ is the temperature.

5 Must Know Facts For Your Next Test

1. Each translational, rotational, or vibrational degree of freedom contributes $\frac{1}{2}k_BT$ to the total energy.
2. Monatomic gases have three translational degrees of freedom, leading to an average kinetic energy of $\frac{3}{2}k_BT$ per atom.
3. Diatomic gases also have rotational degrees of freedom, adding another $k_BT$ to the total energy for a total of $\frac{5}{2}k_BT$ per molecule.
4. The theorem helps explain why heat capacities differ between monatomic and diatomic gases.
5. Equipartition breaks down at very low temperatures where quantum effects become significant.

Review Questions

• What is the average energy contributed by each degree of freedom according to the equipartition theorem?
• How does the equipartition theorem explain the difference in heat capacities between monatomic and diatomic gases?
• Why does the equipartition theorem fail at very low temperatures?

Related terms

Degree of Freedom: An independent mode in which a particle can store energy, such as translational, rotational, or vibrational motion.

Boltzmann Constant ($k_B$): $A fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature; its value is approximately 1.38 \times 10^{-23}$ J/K.

$Heat Capacity: $The amount of heat required to change a substance's temperature by one degree Celsius; it depends on the number and type of degrees of freedom available to particles within that substance.