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Residual entropy has to do with the placement of certain atoms within a molecule. For example, a molecule like CH3Cl would have residual entropy because there are four different places (degeneracy) where you can put the Cl in the tetrahedral shape. A molecule like O2 on the other hand only has one configuration, so there is no residual entropy.
Residual entropy is mostly considered in situations where all types of entropy is removed, such as the case of cooling molecules to almost 0 K. Under these conditions, the particles are considered motionless and thus we ignore aspects like entropy associated with gaseous phase particles, etc.
In class today, we spoke about residual entropy mostly in terms of absolute cooling (where the temperature is 0 K, or rather, as close to 0 K as we can presently reach). In molecules where degeneracy > 1 exist, even in the event that 0 K were reached, some level of residual entropy would exist due to the possible arrangement of atoms.
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