# Statistical Thermodynamics and Rate Theories/Degrees of freedom

Molecular degrees of freedom refer to the number of ways a molecule in the gas phase may move, rotate, or vibrate in space. Three types of degrees of freedom exist, those being translational, rotational, and vibrational. The number of degrees of freedom of each type possessed by a molecule depends on both the number of atoms in the molecule and the geometry of the molecule, with geometry referring to the way in which the atoms are arranged in space. The number of degrees of freedom a molecule possesses plays a role in estimating the values of various thermodynamic variables using the equipartition theorem. These molecular degrees of freedom essentially describe how a molecule is able to contain and distribute its energy.

Degree of freedom Monatomic Linear molecules Non-linear molecules
Translational 3 3 3
Rotational 0 2 3
Vibrational 0 3N - 5 3N - 6
Total 3 3N 3N

## Translational degrees of freedom

Translational degrees of freedom arise from a gas molecule's ability to move freely in space. A molecule may move in the x, y, and z directions of a Cartesian coordinate system, appearing at a new position in space (relative to a starting position) via translation. A gas molecule is not restricted in which direction it may move, thus it has three translational degrees of freedom. This holds true for all gas molecules, whether they are monatomic, diatomic, or polyatomic, as any molecule may move freely in all directions in three-dimensional space.

## Rotational degrees of freedom

A molecule's rotational degrees of freedom represent the number of unique ways the molecule may rotate in space about its center of mass which a change in the molecule's orientation. These axes may pass through atoms or bonds. By examining the molecule's symmetry (and, by extension, its geometry), the number of rotational degrees of freedom can quickly and easily be determined. A monatomic gaseous molecule such as a noble gas possesses no rotational degrees of freedom, as the center of mass sits directly on the atom and no rotation which creates change is possible.

A diatomic molecule, like H2 or HCl, has two rotational degrees of freedom. The center of mass of a linear molecule rests somewhere between the two terminal atoms. In the case of HCl it exists somewhere along the bond. The center of mass can be taken as the origin of a three-dimensional Cartesian grid, the z axis of which runs along the bond and through the two atoms. Rotation about the x and y axes generates a noticeable change in the molecule's orientation, while rotation about the z axis (analogous to the monatomic case) produces no change in the molecule and is considered a 'lost' rotational degree of freedom.

A polyatomic molecule may have either two or three rotational degrees of freedom, depending on the geometry of the molecule. For a linear polyatomic, such as CO2 or C2H2, the molecule has only two rotational degrees of freedom. The reason for this is discussed in the previous paragraph.

## Vibrational degrees of freedom

The number of vibrational degrees of freedom (or vibrational modes) of a molecule is determined by examining the number of unique ways the atoms within the molecule may move relative to one another, such as in bond stretches or bends. This can be determined mathematically using the $3N-6=n_{DOF}$  rule, where $N$  is the number of atoms in the molecule and $n_{DOF}$  is the number of vibrational degrees of freedom. Note that for linear molecules, the rule instead becomes $3N-5=n_{DOF}$ , meaning a linear polyatomic made up of $N$  atoms will have one more vibrational degree of freedom than a non-linear polyatomic with $N$  atoms. Application of the rule for linear molecules shows that a diatomic molecule has only one vibrational degree of freedom. This is a logical conclusion, as there is only one bond vibration possible, a stretch of the bond between the two atoms.

As an example, water, a non-linear triatomic, should have three vibrational degrees of freedom, since $3(3)-6=3$ . The three vibrational modes corresponding to these three vibrational degrees of freedom can be seen above. These vibrational modes were determined computationally.