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 (CH3-CH3) 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 Å2 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 Å2 (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:
|Free internal rotation||7.5||12.0||14.2||17.1|
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 Å2. The moment of inertia for a methyl attached to an infinitely heavy mass will be about 3 amu Å2 . For example, in (E)-1,3-pentadiene the methyl rotor reduced moment of inertia is 2.987 amu Å2
See also the discussion on internal rotation in section I.D. A brief description of the thermochemical quantities and methods.