Equipartition theorem

Energy shared among degrees of freedom

The equipartition theorem is a powerful result from classical statistical mechanics. It says that each independent quadratic degree of freedom contributes an average energy of

rac{1}{2}k_BT

per molecule.

A degree of freedom is an independent way a molecule can store energy. Translational motion in x, y, and z directions gives three degrees of freedom. Molecules may also rotate and vibrate, depending on structure and temperature.

Monatomic ideal gas

A monatomic ideal gas molecule, such as helium or argon, can translate in three dimensions. Its translational kinetic energy is

K=rac{1}{2}mv_x^2+rac{1}{2}mv_y^2+rac{1}{2}mv_z^2

Each term is quadratic and contributes rac{1}{2}k_BT. Therefore the average energy per molecule is

angle=rac{3}{2}k_BT$$ For $N$ molecules, $$U=rac{3}{2}Nk_BT=rac{3}{2}nRT$$ This matches kinetic theory. ## Heat capacity of monatomic gases At constant volume, added heat changes internal energy: $$C_V=left(rac{partial U}{partial T} ight)_V$$ For a monatomic ideal gas, $$C_V=rac{3}{2}nR$$ and $$C_P=C_V+nR=rac{5}{2}nR$$ The heat capacity ratio is $$gamma=rac{C_P}{C_V}=rac{5}{3}$$ ## Diatomic molecules A diatomic molecule, such as nitrogen or oxygen, can translate and rotate. At ordinary temperatures, it has three translational and two rotational quadratic degrees of freedom. This gives approximately $$langle E angle=rac{5}{2}k_BT$$ per molecule, and $$C_Vapproxrac{5}{2}nR$$ Vibrational modes may be inactive at room temperature because quantum energy spacing is too large compared with $k_BT$. At higher temperatures, vibrations can become active and increase heat capacity. ## Vibrational degrees of freedom A vibrational mode has both kinetic and potential energy terms, each quadratic. Therefore one vibrational mode contributes $$k_BT$$ on average: half kinetic and half potential. This is why activating vibrational modes can significantly increase heat capacity. ## Classical limits and quantum corrections Equipartition is classical. It assumes energy can vary continuously. At low temperatures or for modes with large quantum spacing, some degrees of freedom are frozen out and do not contribute the classical amount. This explains why measured heat capacities vary with temperature and why classical equipartition fails for solids at low temperatures and for molecular vibrations at ordinary temperatures. ## Solids and Dulong-Petit law In a simple classical solid, each atom vibrates in three dimensions. Each direction has kinetic and potential quadratic terms, giving six quadratic terms total. The average energy per atom is $$3k_BT$$ For one mole of atoms, $$C_Vapprox 3R$$ This is the Dulong-Petit law, which works well for many solids at high temperature but fails at low temperature due to quantum effects. ## The big idea The equipartition theorem links temperature to average energy in quadratic degrees of freedom. It explains ideal gas internal energy and heat capacities, predicts contributions from translation, rotation, and vibration, and reveals where quantum physics must replace classical reasoning.