Entropy and a box of air

As an example of entropy in action, consider the air in the building in which you're sitting. You assume the air is evenly distributed throughout it. If you go to the kitchen for a beverage, you don't stop to wonder "Gee, will there be air in the kitchen when I get there?" Why don't we have to worry about this?

The answer (as you've already guessed if you're on the ball) is entropy.

To see why, let's consider a box of air, with only 10 molecules in it.

I've divided the box in half with a dashed line. The molecules in this box will be zipping around very fast, bouncing off the walls, off each other, and so forth. As a result, they'll be distributed randomly throughout the box. Right now, the number of molecules in the left half of the box is about the same as in the right half. But it doesn't have to be that way.

Here, 9 of the 10 molecules are on the left side of the box. What are the odds of that happening? About one in 102. (Where did that number come from?) On the other hand, the odds of half the molecules being on each side is about one in 4. We can express this in terms of microstates and macrostates. The macrostate with 9 molecules in the left half has 10 microstates (each of which has a different molecule on the right). The macrostate with 5 molecules on each side has 252 microstates. In other words, the macrostate with 5 molecules on each side has a higher entropy than the one with 9 molecules on the left. And since all microstates are equally likely (each molecule has the same chance of being on the left or the right) a macrostate with higher entropy (more microstates) is more likely than one with lower entropy. In the case of the two macrostates shown above, the one with the even distribution is about 25 times as likely (252/10) as the one with the uneven distribution.

As we consider boxes with more particles, the preference for an even distribution gets much much stronger as the number of microstates in a macrostate explodes. For example, suppose we have a box with 1000 molecules. The number of microstates with 500 molecules on each side is about 10^299 (that is, 1 followed by 299 zeroes) while the number of microstates with 900 molecules on one side of the box is only about 10^140. So the former macrostate has a much larger entropy than the latter, and is about 10^159 times as likely. And if that seems like an astronomical number, keep in mind that when it comes to the number of molecules in a gas, 1000 doesn't even begin to describe it. The number of air molecules in a shoebox is around 10^22. So when we have a house full of air, we can expect the molecules to distribute themselves pretty evenly throughout. Such a state has a much greater entropy, and so is much more likely to occur. So go ahead wander into the kitchen to get yourself a cold one. There'll be plenty of air when you get there. Probably.

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Here's where that number (and some of the others) came from. Each molecule can be in either the left or the right half. That's two possibilities, and with 10 particles the total number of possibilities is 2^10 = 1024. Now of those 1024 possibilities, 10 of them have 9 molecules on the left. (The one on the right can be any one of the 10 molecules.) So the odds of having 9 molecules on the left is 10/1024, or about 1 in 102.

The same idea lies behind the other numbers, but it takes factorials to actually calculate them. For example, to calculate the number of microstates with 90 molecules on one side and 10 on the other, we calculate 1000!/(900! 100!) =6.4 x 10^139. If you've ever played with factorials before, you know they get very big very fast. So when I say that one macrostate is more likely than another, I mean really really more likely!

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