1. What the Law of Equipartition Says
The law of equipartition of energy states that energy is shared equally among all active degrees of freedom of a molecule. Each degree of freedom contributes an average energy of:
\( \dfrac{1}{2} k_B T \)
where:
- \( k_B \) is the Boltzmann constant
- \( T \) is the absolute temperature (in Kelvin)
2. Energy Contribution per Degree of Freedom
For each active degree of freedom, the molecule has:
E_{per\ freedom} = \dfrac{1}{2} k_B T
If a molecule has f degrees of freedom, then the average energy of one molecule becomes:
E = \dfrac{f}{2} k_B T
2.1. Example
If a molecule has 3 degrees of freedom (like a monatomic gas), it has:
E = \dfrac{3}{2} k_B T
3. Equipartition for Different Types of Gases
Because different gases have different degrees of freedom, the energy contribution varies.
3.1. Monatomic Gas (f = 3)
Only translational motion is present.
E = \dfrac{3}{2} k_B T
3.2. Diatomic Gas (f = 5 at normal temperature)
Includes 3 translational + 2 rotational degrees of freedom.
E = \dfrac{5}{2} k_B T
3.3. Polyatomic Gas (f = 6)
3 translational + 3 rotational degrees of freedom.
E = 3 k_B T
4. Relation to Molar Heat Capacities
The law of equipartition helps explain why gases have different heat capacities. Using:
E = \dfrac{f}{2} k_B T
the energy per mole becomes:
E_{mole} = \dfrac{f}{2} R T
(where R is the gas constant).
4.1. Heat Capacity at Constant Volume
C_V = \dfrac{f}{2} R
4.2. Heat Capacity at Constant Pressure
C_P = C_V + R
5. Why Some Degrees of Freedom Are Inactive
Rotational and vibrational modes in molecules do not always contribute energy. For example, in diatomic gases:
- Vibrational modes require high energy.
- At room temperature, vibrational modes are usually inactive.
5.1. Effect of Temperature
At higher temperatures, vibrational degrees of freedom become active, increasing heat capacity. This explains why the heat capacity of gases increases with temperature.
6. Simple Everyday Interpretation
If more ways to store energy are available (more degrees of freedom), more heat is needed to raise the temperature by the same amount. This explains why:
- Monoatomic gases heat up quickly
- Diatomic gases heat more slowly
- Polyatomic gases heat even more slowly