## VII.C.2. |

# How much does a methyl rotor (internal rotation) contribute to the entropy?

The contribution to the entropy for a methyl rotor depends on the reduced moment of inertia and the barrier to internal rotation.

Using ethane (CH_{3}-CH

_{3}) as an example we can compute the contribution to entropy of the methyl internal rotation. We will use results calculated at HF/6-31G* and a temperature of 298.15K and determine the contribution to the entropy three different ways:

1. Treating the methyl internal rotation as a harmonic oscillator we have a scaled frequency of 292.6 cm

^{-1}. This leads to a contribution to the entropy of 6.1 J K

^{-1}mol

^{-1}.

2. Treating the methyl rotation as a free internal rotor with a reduced moment of inertia of 1.55 amu Å

^{2}leads to a contribution to the entropy of 12.0 J K

^{-1}mol

^{-1}.

3. The correct treatment as a hindered rotor with a finite barrier of 1044 cm

^{-1}and a reduced moment of inertia of 1.55 amu Å

^{2}(both from a HF/6-31G* calculation) yields 6.8 J K

^{-1}mol

^{-1}. This can be summarized for several temperatures in the following table:

100K | 298.15K | 500K | 1000K | |
---|---|---|---|---|

Harmonic oscillator | 0.7 | 6.1 | 10.0 | 15.6 |

Free internal rotation | 7.5 | 12.0 | 14.2 | 17.1 |

Hindered rotor | 0.7 | 6.8 | 11.2 | 16.1 |

At low temperatures the harmonic oscillator is a good approximation, and at very high temperatures the free rotor is a better approximation.

The reduced moment of inertia for a methyl group varies
depending on the rest of the molecule. In the ethane example it is 1.55 amu
Å^{2}. The moment of inertia for a methyl attached to an infinitely
heavy mass will be about 3 amu Å^{2} . For example, in
(E)-1,3-pentadiene the methyl rotor reduced moment of inertia is 2.987 amu
Å^{2}

See also the discussion on internal rotation in section I.D. A brief description of the thermochemical quantities and methods.