Solution to Exercise 2.1-2"Do the Math for the Formation Entropy"

We start with
Next we must do the differentiation, i.e. form F/T:
One can go straight ahead, of course. But here comes a little helpful trick: Multiply skillfully by T/T and re-sort; you get
Now we need to resort to approximations
First we realize that whenever h ·w/2p << kT, then
This takes care of the first term.
The second term needs a somewhat more sophisticated approach. Substituting x for h · w/2p · kT, we can use a simple expansion formula, stop after the second term and re-insert the result. This gives
That's as far as one can go. Now use w' for the circle frequencies of the crystal with a vacancy and form SF = S'S

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