## Perfect Crystal

Boltzmann Equation for Entropy: $S = k_{B} \ln W$

Crystal Ma
Posts: 9
Joined: Fri Sep 25, 2015 3:00 am

### Perfect Crystal

What is the definition for perfect crystal? and what does professor mean by that there is no impurity and defect?

Also, According to the formula, $S=K_{B}LnW$, should the entropy be 0 if W=1, even when the temperature is not 0 degree?

hpage204
Posts: 19
Joined: Fri Sep 25, 2015 3:00 am

### Re: Perfect Crystal

A perfect crystal is a "perfectly ordered substance". In other words, it is a sample of something in which all the molecules are lined up perfectly and there are no imperfections. I think it's easier to understand when you consider $S=K_{b}lnW$. Think about calculating degeneracy for a substance like BH2F, for example. W=3 because there are three possible arrangements of the molecule, one for each possible location of the fluorine atom. If it were BH3, W would be 1 because, no matter how it's arranged, Boron would be surrounded by three hydrogens. Now consider this as if the molecule were a crystal (this is just a metaphor, a single molecule couldn't be a perfect crystal I think). The flourine could be viewed as an imperfection. Because there are so many different arrangements for the imperfection to be in many different positions, the molecule has a high degeneracy. If there were no imperfections (similar to BH3), no matter how the crystal is arranged, everything is in line perfectly. An imperfection in a crystal could be one atom of a different element in the middle of a crystal consisting of some other substance or simply one particle being pulled out of whack. If there are no imperfections and everything is at 0K, then there is no disorder. Nothing is moving and the arrangement cannot be changed.

As for your other question, yes, if W=1 then the residual entropy is zero (when we use $S=K_{b}*lnW$, we're calculating the residual entropy). If you look at Q3A from the Winter 2012 midterm, for example, the residual entropy of O2 gas is zero because W=1. This makes sense because residual entropy is basically the entropy the substance has when T=absolute zero, it's often described as "the difference between the calculated value of entropy and the experimentally determined value of entropy for a system". The only way we can calculate the total entropy is if we assume that the system is a perfect crystal at zero kelvins, but experiments have shown that isn't the case. Residual entropy tells us what the difference is. Because O2 could form a perfect crystal and degeneracy is 1, the difference between the calculated and empirical values is basically zero, and the residual entropy is zero.

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