Equipartition of Energy

Number of Degrees of Freedom- total number of ways of absorbing energy. Equal to 3 N where N is the number of atoms in the molecule. (Equal to number of coordinates for the system. For two atoms you would have x,y,z for each atom. Simpler coordinate systems are available for connected atoms.)

Nuclear Motions of Polyatomic molecules.

Translational - movement in the x,y,z position of the molecule-always 3 and only 3 of these for any gaseous molecule.

Rotational- rotation of the molecule around the x, y, or z axis. Either 2 or 3 depending on whether the molecule is linear or non linear.

Vibrational- stretching or bending of bonds. The rest of the motions are classified as these so that #vibrations= 3N-(5 or 6) depending on whether the molecule is linear or non-linear.

Examples:

1. NH3 Degrees of freedom = 12

3 translational 3 rotational 6 vibrational

2. HCl Degrees of freedom = 6

3 translational 2 rotational 1 vibrational

Equipartition of energy- each way the molecule has to absorb energy contributes R/2 to Cv at high temperatures.

For gases:

Each translational degree of freedom contributes R/2 to heat capacity, Cv (RT/2 to energy).

Each rotational degree of freedom contributes R/2 to heat capacity, Cv. (RT/2 to energy).

Vibration has both a potential and a kinetic energy component. Because of this each will contribute R to Cv at high temperatures, but because the vibrations are excited only at temperatures much greater than room temperature the usual assumption is that the contribute 0.

For solids.

Only vibrations are present.

Metallic vibrations are fully excited at room temperature and ionic solids are not.

Predict the heat capacities, Cv , of the following: