Entropy 

Entropy, always abbreviated by a capital "S", measures the degree of disorder in a system in a quantitative way. It thus comes up with a number for disorder.  
How can you put a number on the
disorder in your daughters room? You certainly know the room is disorderly (and that she doesn't
have that from you), and you can assess it qualitatively on a scale going from "almost
good" via "clean up or else" to "I'm now going to see my
lawyer to strike you from my will". You make assessments like that all the
time, but how can you come up with a number? Well, to be honest, for complex systems like your daughter's room, you could come up with a number in principle but in reality you cannot. But for much simpler systems, like a large bunch of iron atoms, it is perfectly possible to find definitions that can and will give you a meaningful number. 

The clue for putting a number on disorder lies in looking first at the ageold definition of order:  


What this means is that there is only one possibility to have order in a system; there is only one way to arrange things orderly. Here is a picture of the opposite:  


Perfect order implies that the white color is in the can and nowhere else. Two happy kids have found a way to distribute a lot of paint in many places where paint does not belong. Disorder in the room has most definitely increased. Note also that order is boring.  
Let's look at two model systems to
grasp how we define disorder now.


Now ask yourself: In how many ways can I distribute my n_{So} dirty socks on the available N_{room} places? My n_{V} vacancies on the available N_{Crystal} places?  
From the way I set this up you probably figure that these two questions have the same general answer. Not really. Figure again!  
Allright, I give you a hint. In your bedroom, the
red sock might be under the bed and the blue sock in the wrong drawer. Or it
could be the other way around. You definitely can distinguish these two
arrangements (on top of wherever the other socks are), and you must count it as
two possibilities. However, you cannot distinguish if one vacancy sits on place 15 and another one on place 324, or if it is the other way around. Both arrangements are identical in this case. Oh f...! This is getting messy. It gets even messier if you consider all the possibilities of being able / unable to distinguish between elements / arrangements. For the brave: here is the link; the faint of heart need not apply. 
Don't worry, be happy! You certainly don't want to get immersed into that kind of detail, and the good news is that it is not necessary. All you need to know is that there are unambiguous answers to those questions. There is a definite number of possibilities to distribute n_{So} dirty socks on the available N_{room} places, or n_{V} vacancies on the available N_{Crystal}, or God knows what else. Even better, there are some nerds out there who can calculate those numbers; they have probably nothing better to do.  
Let's use some ageold magic and deal with this
vaguely frightening stuff by giving it a
name. Let's
call the number of possibilities to distribute whatever we want to distribute
on the places available: P, short for
possibilities. There. Now we
can talk about it without really knowing what it is we talk about. If P is large, the disorder is large. Remember: Perfect order means P = 1, there is only one possibility to put things away in perfect order. Now comes major magic: The entropy of a disordered state with P possibilities to create the disorder is 



That is
Boltzmann's famous
equation for the entropy of a system. In words: The entropy S is proportional to the natural logarithm of the number of possibilities P to arrange the things in the system we are looking at. The proportionality constant is Boltzmann's constant k. 

The letter "k" always denotes
Boltzmann's constant: R = 8,314 472 J · mol^{–1} · K^{–1} is the "universal gas constant" and A mol of a substance is defined as that amount of a substance that contains N_{A} atoms or molecules. You always get 1 mol if you take as many grams of a substance as the number giving the atomic weight of its particles (H = 1, C = 12, Si = 28,09, Fe = 55,85, and so on). If it doesn't come back to you now, because you either never learned about the very basics of the world around you or forgot it, I feel sorry for you. More than that I can't do. Some are born to be wild, some are born to be Materials Scientists, but none are born to remain stupid. Get to it! 

Boltzmann's constant is one of the few truly fundamental and universal constants. No theory exists that can calculate it from something more fundamental; we have to measure the numbers for those constants. Other fundamental constants are, for example, the speed of light c, and Planck's constant h.  
Boltzmann's entropy law S = k ·
lnP is just as fundamental and important as Einstein's famous


So if you have P possibilities for arranging the elements of your system, you know the degree of disorder. Take the natural logarithm of that number and multiply by the Boltzmann constant.  
Note that the number you get increases only
slowly with P:


We are done. We can calculate the entropy of a system with a bit of combinatorics. Since we also can calculate the energy, we can combine the two in the free energy and use that to calculate the nirvana conditions of the system.  
If that doesn't sound like something you want to spend your free time with, you are a guy like me. A bit lazy, maybe, but not crazy. Thank Boltzmann (and others), we don't need to go through all that steps. They have done that for us in full generality once and for all, coming up with ingenious ways to get what we want in a far simpler if more abstract ways. This module gives a glimpse  
Now that we can calculate the entropy
of some system, we can go places. For example to:


Group 16 / VIA; Chalkogenides or Oxygen Group
The Second Law and Computer Science
Units of Length, Area, and Volume
Early Metal Technology  1. Gold
© H. Föll (Iron, Steel and Swords script)