## Equi-partition of energy

Teaching Guidance for 14-16

Statistical studies, combined with the assumption that every molecular collision is elastic, lead to the conclusion that the molecules of all gases at the same temperature store the same amount of energy kinetically. This ignores one very important physics principle, the equi-partition of energy theorem.

The full form of this theorem states that each degree of freedom will on average have the same energy. The linear motion of molecules entails three degrees of freedom – motions in the *x*, *y* and *z* directions. A gas whose molecules are simple atoms, such as helium or neon, has only these three kinds of motion. A molecule made from two or more atoms, however, can move in additional ways: it will have rotations and vibrations.

At the beginning of the twentieth century, it seemed clear that equi-partition should apply to the energies of rotation and vibration. However, certain experiments, such as those measuring thermal capacities over a wide range of temperatures, threw increasing doubt on this. The full form of equi-partition theorem failed, and quantum theory explained why.

Equi-partition among linear motions still applies. All gas molecules, at a given temperature, have the same average kinetic energy.

#### Useful results

Average energy for molecules of gas A = average energy of molecules for molecules of gas B, at the same temperature.
*m*_{A} < *v*_{A}^{2}> = *m*_{B} < *v*_{B}^{2}>

So you can compare molecular speeds if you know the relative molecular masses.

Another equation derived from the kinetic theory of gases:
*pV* = *1/3 Nm* < *c*^{ 2} > where *p* is the gas pressure, *V* is gas volume, *N* the number of molecules, *m* their mass and < *c*^{ 2} > their mean square speed. This equation tells you that equal volumes of gases at the same pressure and temperature must contain the same number of molecules (Avogadro’s rule).