4  Low-Temperature Entropy (Alleged Third Law)

As mentioned in the introduction, one sometimes hears the assertion that the entropy of a system must go to zero as the temperature goes to zero.

There is no theoretical basis for this assertion, so far as I know – just unsubstantiated opinion.

As for experimental evidence, I know of only one case where (if I work hard enough) I can make this statement true, while there are innumerable cases where it is not true:

• There is such a thing as a spin glass. It is a solid, with a spin at every site. At low temperatures, these spins are not lined up; they are highly disordered. And there is a large potential barrier that prevents the spins from flipping. So for all practical purposes, the entropy of these spins is frozen in. The molar entropy involved is substantial, on the order of one J/K/mole. You can calculate the amount of entropy based on measurements of the magnetic properties.
• A chunk of ordinary glass (e.g. window glass) has a considerable amount of frozen-in entropy, due to the disorderly spatial arrangement of the glass molecules. That is, glass is not a perfect crystal. Again, the molar entropy is quite substantial. It can be measured by X-ray scattering and neutron scattering experiments.
• For that matter, it is proverbial that perfect crystals do not occur in nature. This is because it is energetically more favorable for a crystal to grow at a dislocation. Furthermore, the materials from which the crystal was grown will have chemical impurities, not to mention a mixture of isotopes. So any real crystal will have frozen-in nonuniformities. The molar entropy might be rather less than one J/K/mole, but it won’t be zero.
• If I wanted to create a sample where the entropy went to zero in the limit of zero temperature, I would proceed as follows: Start with a sample of helium. Cool it to some very low temperature. The superfluid fraction is a single quantum state, so it has zero entropy. But the sample as a whole still has nonzero entropy, because ³He is quite soluble in ⁴He (about 6% at zero temperature), and there will always be some ³He around. To get rid of that, pump the sample through a superleak, so the ³He is left behind. (Call it reverse osmosis if you like.) Repeat this as a function of T. As T goes to zero, the superfluid fraction goes to 100% (i.e. the normal-fluid fraction goes to 0%) so the entropy, as far as I know, would go to zero asymptotically.

Note: It is hard to measure the low-temperature entropy by means of elementary thermal measurements, because typically such measurements are insensitive to “spectator entropy” as discussed in section 12.6. So for typical classical thermodynamic purposes, it doesn’t matter whether the entropy goes to zero or not.