therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1 and wavelengths of approximately 30 to 3 μm. Vibrations of polyatomic molecules are desc For a molecule with N atoms, the positions of all N nuclei depend on a total of 3N coordinates, so that the molecule has 3N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates. A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates. An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third rotational coordinate. The number of vibrational modes is therefore 3N minus the number of translational and rotational degrees of freedom, or 3N − 5 for linear and 3N − 6 for nonlinear molecules. sy…

3N is most often referred to as the total number of degrees of freedom of the molecule being investigated. The total number of degrees of freedom, can be divided into: - 3 coordinates to describe the translational motion around the center of mass; these coordinates are called the translational degrees of freedom - 3 coordinates to describe the rotational motion in non-linear molecules; for linear molecules only 2 coordinates are required; these coordinates are called the rotational degrees of freedom - the remaining coordinates are used to describe vibrational motion; a non-linear molecule has 3N - 6 vibrational degrees of freedom whereas a linear molecule has 3N -5 degrees of freedom. | Table 1: Overview of degrees of freedom | | | | | |---|---|---|---|---| | | Total Degree of Freedom | Translational degrees of freedom | Rotational degrees of freedom | Vibrational degrees of freedom | |---|---|---|---|---| | Nonlinear Molecules | 3N | 3 | 3 | 3N -6 | | Linear Molecules | 3N | 3 | 2 | 3N - 5 | Ethane, \(C_2H_6\) has eight atoms (\N=8\) and is a nonlinear molecule so of the \(3N=24\) degrees of freedom, three are translational and three are rotational.