Third law of thermodynamics and entropy Third law of thermodynamics states: "The entropy of a perfect crystal at absolute zero is exactly equal to zero". My question is why entropy at such state is $0$ ? Let say my crystal have $n$ molecules/particles, isn't third law assuming that those particels are indistinguishable? So $n$ particles at $T = 0$ form same pattern but what will happen if for me as some observer (if I could trace every single partice) it make difference whare each particular particle is positioned in pattern?
 A: The term to look up is residual entropy.
If you build a solid out of a bunch of distinguishable particles, it will indeed have nonzero residual entropy because of that. Why does the third law assume that atoms of the same type are truly indistinguishable? Because that assumption is correct†!! At least in our universe!
As for the statement "The entropy of a perfect crystal at absolute zero is exactly equal to zero", that's fine, but note that there is sort of a terminology issue here. For example, consider proton disorder in (some types of) water ice. The protons (a.k.a. hydrogen nuclei) are displaced from the symmetric position in the crystal lattice, in an irregular way, and that leads to residual entropy. I guess the question is: Does this kind of ice really count as a "crystal"? If the protons are offset in an irregular way, then the configuration no longer has translational symmetry, so according to the technical definition of a crystal, this kind of ice would not qualify as a crystal. But in an everyday sense, we do say that ice is a crystal. So be aware of that.
†Isotopes are an interesting case. A crystal of "pure" silicon actually consists of a mixture of two or three distinguishable nuclei (Si-28, Si-29, Si-30). Technically, this crystal has a residual entropy from this, but we usually ignore it because it has no practical relevance in everyday chemistry. (I've never seen a discussion of isotopic mixing entropy, so I'm guessing a bit here...) I think that textbook writers implicitly take the "entropy of mixing" for the natural isotopic mixture of every element, and the entropy numbers we tabulate and calculate are actually the entropy numbers beyond that baseline. Again, you can use terminology how you like. If you define the word "crystal" to imply isotopic purity by definition, then you can say that no crystal has any residual entropy of any sort. But this is clearly not how anyone uses the term "crystal" in practice.
A: The entropy is zero because if you change the state of any one or more particles it will no longer have the same total energy. 
