# Statistical Thermodynamics and Rate Theories/Degrees of freedom

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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 determining the values of various thermodynamic variables using the equipartition theorem.

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 an axis. This axis 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 it is most simply represented as a sphere in space. It is thus uniform on all sides, meaning a rotation produces no visible change in the molecule's appearance or orientation in space.

A diatomic molecule, like H2 or HCl, has two rotational degrees of freedom. If the bond between the two atoms is taken to be the z-axis of a three-dimensional cartesian coordinate system, then the molecule can rotate within both the xz and yz planes and produce a visible change. If it were to rotate within the xy plane (around the z axis), no change would be observed, thus limiting the number of rotational degrees of freedom to two.

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 the same as for a diatomic molecule: there is no change in the molecule's apparent orientation when rotated along the C axis. A non-linear polyatomic, like H2O or NH3, has three rotational degrees of freedom, as a rotation along any of the Cartesian axes changes the molecule's orientation and appearance in space (from a stationary viewpoint).

## 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 less 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.