# Greater entropy: an atom or macromolecule?

A question that appeared on my last chemistry exam was :

Which of the following has greater entropy A) An atom B) A macromolecule

The question doesn't specify anything else(i.e. type/size of atom or macromolecule). The macromolecules we learned class were proteins, sugars, nucleic acids(DNA/RNA) or lipids. What is the correct answer and why(hypothetical calculation, if possible) ?

There are many equations that describe the entropy of a system, one is $$\ce{S = k~ ln ~W}$$ where S is the entropy of the system, k is Boltzmann's constant and W is the number of states a system can exist in. If we take an atom, it can only move in 3 directions x, y and z - that's it. If we take a small molecule like ethane, it can also move in these same 3 dimensions but it can also rotate about any one of the atoms, the various bonds can stretch and compress and the various bond angles can expand and collapse. Clearly, there are a lot more states that ethane can exist in as compared to a single atom, clearly W is larger for ethane than a single atom, clearly S is larger for ethane than a single atom. As we increase the complexity of the molecule, W and S increase exponentially. A macromolecule will have a much larger entropy than an atom.

Edit (based on @Gdgames Gamers comment below)

Entropy is a measure of the disorder in a system. It will always be true that the entropy of substance in the gas phase will always be greater than the entropy of the same substance in the liquid phase, which, in turn, will be always greater than the entropy of the same substance in the solid phase.

Perhaps the person who wrote the exam is assuming that the atom is in the gas phase and the macromolecule is in the solid phase and that based on the phase difference alone the atom has a larger entropy than the macromolecule. I don't think this is always correct. First off, the temperature is not specified, nor the state of the macromolecule (is it in solution?). The macromolecule, depending upon the temperature and state, may or may not translate and rotate through space, but it will vibrate (even at 0K). I'm just not sure how to compare the entropy contribution from the potential 3n-6 vibrational modes in the macromolecule to the entropy from the 3 translational modes of the atom. Very poor question by the examiner.

• While you are correct, I can't help but feel the question was posed incorrectly in the first place. Knowing that a collection of atoms has more entropy than a single atom feels like it yields no information. A much more interesting question would have been to compare the entropy per atom (or per mole of atoms) when free and when bound in a macromolecule. Of course, the macromolecule has much lower entropy than the sum of its separate constituents. Jul 19 '14 at 15:46
• @Nicolau Saker Neto If we have a non-linear molecule of n atoms it will have 3 translational, 3 rotational and 3n-6 vibrational degrees of freedom, e.g. it will have 3n degrees of freedom. n independent atoms would also have 3n degrees of freedom. So wouldn't the entropy per atom of n independent atoms be equal to the entropy per atom of n connected atoms?
– ron
Jul 19 '14 at 16:07
• I don't think calculating degrees of freedom alone is enough to evaluate the entropy of a system. A macromolecule can store entropy in vibrational degrees of freedom, which a gas of free atoms cannot, but there is very little vibrational excitation at reasonable temperatures, and I think every macromolecule decomposes well before most of its vibrational degrees of freedom can be explored. Jul 19 '14 at 16:50
• @NicolauSakerNeto I thought the question was lacking also...but this is how it appeared on the exam. I picked the answer to be macro molecule but it was marked wrong.. Jul 19 '14 at 17:15