# Absolute entropy

The third law of thermodynamics states that the absolute entropy of a pure crystalline substance is $0$ at $0\,\mathrm{K}$. In the analysis of chemical reactions this absolute scale is sued to calculate the entropy change. However, it is also noted that some substances like $\text{C}{\text{O}_{2}}$ have residual entropy at $0\,\mathrm{K}$ because their structure is not perfectly crystalline.

Now my question is, given all of the above why is it that in thermodynamics tables such as in the book "Physical Chemistry" by Bawendi the entropy of $\text{C}{\text{O}_{2}}$ is listed as $0$ at $0\,\mathrm{K}$ when it should be some finite (albeit small) number? The same goes for other substances like $\text{H}_{2}\text{O}$ whose entropy is listed as $0$ at $0\,\mathrm{K}$ when it should be a finite number.

I understand that different tables use different reference scales and the entropy of $\text{C}\text{O}_{2}$ could be arbitrarily taken as $0$ at $0\,\mathrm{K}$. However the problem is then if the tables give the entropy of a substance relative to the entropy of the same substance at $0\,\mathrm{K}$, then the reference scale for each substance is different. This is fine but in all the example problems in the textbook mentioned above (and others) they use the entropy values listed in the table for each substance without accounting for the different reference states.

For example, if I have the reaction$$\text{A}~+~\text{B} ~~\longrightarrow ~~ \text{AB} \,,$$the entropy change is$$\Delta S ~=~S_{\text{AB}} - S_{\text{A}} - S_{\text{B}} \,,$$where $S_{\text{A}}$ is the entropy of $\text{A}$ and so on.

Now in order for this to work the entropies must be taken relative to the same reference state (like when we use heat of formation to calculate enthalpies). However in the Bawendi book they use the values listed in the tables for each substance, which are again relative to the entropy at $0\,\mathrm{K}$ of that particular substance. This would all be fine if the entropy at $0\,\mathrm{K}$ was indeed $0$ for all substances but it is not. Could someone please explain why a lot of books do this? or is there something that I am missing?

## 2 Answers

First of all, entropy is an absolute value; you don't use different reference scales and reference states for it. Internal energy is another story: it never appears in the equations as an absolute value, but only as a difference of starting and ending values, hence you may define a scale for it and place zero on that scale more or less arbitrarily. Ditto for enthalpy and Gibbs energy. But entropy is not like that.

Now to the point. The only thing that has zero entropy at $0\,\mathrm{K}$ is an ideal crystal, which is also the equilibrium state (whatever that might mean) of crystalline compounds at that temperature. Getting to equilibrium in fluids might be anything from slow to very fast; in solids it is usually rather slow, and in solids near absolute zero it is so slow that the Sun might well become a red giant before the last defects leave your real crystal. See, thermodynamics is all about eternity. Wait until then, and you'll see the Third law at work.

That's what we mean when we are talking of residual entropy.

I know about the possible degeneracy of the ground state, but this post is already getting too long.

Does the said law have no bearing on earthly things, then? Why, it does. Many crystals just get good enough while growing at reasonable temperature. They have very few defects, hence their entropy is already pretty low and gets lower as we cool them down. On the other hand, many other crystals are not that good.

So why would the book list entropy of all compounds at $0\,\mathrm{K}$ as zero? Well, getting to absolute zero is infinitely hard, and getting close to it is pretty hard too. Maybe the authors just put all those zeros there in a well-intentioned attempt to remind the readers of the Third law? Too bad for those who by that point have heard of residual entropy.

Then again, I don't see why would $\ce{CO2}$ have much of residual entropy in the first place. It is a good rigid molecule with nice polar bonds and without nasty things like nuclear spin. Perhaps its entropy near the absolute zero is really $0.0$ within the given precision. Water ice is another story, with all its hydrogen disorder. But wait, isn't it transforming into a hydrogen-ordered form at low temperatures? So maybe the zeros are true, after all?

The crystals do have to be perfect, any old crystal at $0$ K will not have zero entropy, and so $\ce{CO2}$ and $\ce{H2O}$ are given zero entropy. On the other hand, CO has the possibility of having arrangements CO CO or OC CO so should not have zero entropy at zero K.

The convention usually used in Chemistry is therefore to make the entropy zero when translational, configurational, rotational, vibrational and electronic contributions are zero. Contributions due to nuclear spin and isotope mixing are ignored. This can be justified because nuclei are conserved in reactions and isotopic ratios are effectively constant thus any changes cancel out when taking entropy differences.