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# 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:
Contribution to Entropy for different treatments of the internal rotation in Ethane
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 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.