# 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
|  | q.e.d.

With frame

Do the Math for the Formation Entropy

© H. Föll (Defects - Script)