degrees of freedom

1. The number of independent parameters required to specify the configuration of a system. This concept is applied in the kinetic theory to specify the number of independent ways in which an atom or molecule can take up energy. There are however various sets of parameters that may be chosen, and the details of the consequent theory vary with the choice. For example, in a monatomic gas each atom may be allotted three degrees of freedom, corresponding to the three coordinates in space required to specify its position. The mean energy per atom for each degree of freedom is the same, according to the principle of the equipartition of energy, and is equal to kT/2 for each degree of freedom (where k is the Boltzmann constant and T is the thermodynamic temperature). Thus for a monatomic gas the total molar energy is 3LkT/2, where L is the Avogadro constant (the number of atoms per mole). As k = R/L, where R is the molar gas constant, the total molar energy is 3RT/2.

In a diatomic gas the two atoms require six coordinates between them, giving six degrees of freedom. Commonly these are interpreted as six independent ways of storing energy: on this basis the molecule has three degrees of freedom for different directions of translational motion, and in addition there are two degrees of freedom for rotation of the molecular axis and one vibrational degree of freedom along the bond between the atoms. The rotational degrees of freedom each contribute their share, kT/2, to the total energy; similarly the vibrational degree of freedom has an equal share of kinetic energy and must on average have as much potential energy. The total energy per molecule for a diatomic gas is therefore 3kT/2 (for translational energy of the whole molecule) plus 2kT/2 (for rotational energy) plus 2kT/2 (for vibrational energy), i.e. a total of 7kT/2. 2. The least number of independent variables required to define the state of a system in the phase rule. In this sense a gas has two degrees of freedom (e.g. temperature and pressure).