Understanding Entropy: Thermodynamics and the Second Law Explained

I've now supplied you with two definitions of the state variable entropy. And it's S for entropy.

The thermodynamic definition said that the change in entropy is equal to the heat added to the system divided by the temperature at which the heat is added.

So obviously, if the temperature is changing while we add the heat, which is normally the case, we're going to have to do a little bit of calculus.

And then you can view this as the mathematical, or the statistical, or the combinatorical definition of entropy.

And this essentially says that entropy is equal to some constant times the natural log of the number of states the system can take on.

And this is the case when all the states are equally probable, which is a pretty good assumption. If you have just a gazillion molecules that could have a

gazillion gazillion states, you can assume they're all roughly equally likely. There's a slightly more involved definition if they

had different probabilities, but we won't worry about that now. So given that we've seen these two definitions, it's a good

time to introduce you to the second law of thermodynamics. And that's this. And it's a pretty simple law, but it explains a whole range

of phenomena. It tells us that the change in entropy for the universe when any process is undergone is always greater

than or equal to 0. So that tells us that when anything ever happens in the universe, the net effect is that there's more entropy in

the universe itself. And this seems very deep, and it actually is. So let's see if we can apply it to see why it explains, or

why it makes sense, relative to some examples. So let's say I have two reservoirs that are in contact with each other.

So I have T1. And let's call this our hot reservoir. And then I have T2.

I'll call this our cold reservoir. Well, we know from experience. What happens if I put a hot cup of water, and it's sharing

a wall with a cold glass of water, or cold cube of water, what happens? Well, their temperatures equalize.

If these are the same substance, we'll end up roughly in between, if they're in the same phase. So essentially, we have a transfer of heat from the

hotter substance to the colder substance. So we have some heat, Q, that goes from the hotter substance to the colder substance.

You don't see, in everyday reality, heat going from a colder substance to a hotter substance. If I put an ice cube in, let's say, some hot tea, you don't

see the ice cube getting colder and the hot tea getting hotter. You see them both getting to some equal temperature, which

essentially the tea is giving heat to the ice cube. Now in this situation there are reservoirs, so I'm assuming that their temperatures stay constant.

Which would only be the case if they were both infinite, which we know doesn't exist in the real world. In the real world, T1's temperature as it gave heat

would go down, and T2's temperature would go up. But let's just see whether the second law of thermodynamics says that this should happen.

So what's happening here? What's the net change in entropy for T1? So the second law of thermodynamics says that the

change in entropy for the universe is greater than 0. But in this case, that's equal to the change in entropy for T1 plus the change in entropy for-- oh, I shouldn't--

instead of T1, let me call it just 1. For system 1, that's this hot system up here, plus the change in entropy for system 2.

So what's the change in entropy for system 1? It loses Q1 at a high temperature. So this equals minus the heat given to the system is Q over

some hot temperature T1. And then we have the heat being added to the system T2. So plus Q over T2.

This is the change in entropy for the system 2, right? This guy loses the heat, and is at temperature 1, which is a higher temperature.

This guy gains the heat, and he is at a temperature 2, which is a colder temperature. Now, is this going to be greater than 0?

Let's think about it a little bit. If I divide-- let me rewrite this. So I can rearrange them, so that we can write this as Q

over T2 minus this one. I'm just rearranging it. Minus Q over T1.

Now, which number is bigger? T2 to T1? Well, T1 is bigger, right?

This is bigger. Now, if I have a bigger number, bigger than this-- when we use the word bigger, you have to

compare it to something. Now, T1 is bigger than this. We have the same number in the numerator

in both cases, right? So if I take, let's say, 1 over some, let's say, 1/2 minus 1/3, we're going to be bigger than 0.

This is a larger number than this number, because this has a bigger denominator. You're dividing by a larger number.

That's a good way to think about it. You're dividing this Q by some number here to get something, and then you're subtracting this Q

divided by a larger number. So this fraction is going to be a smaller absolute number. So this is going to be greater than 0.

So that tells us the second law of thermodynamics, it verifies this observation we see in the real world, that heat will flow from the hot body to the cold body.

Now, you might say, hey, Sal. I have a case that will show you that you are wrong. You could say, look.

If I put an air conditioner in a room-- Let's say this is the room, and this is outside. You'll say, look what the air conditioner does.

The room is already cold, and outside is already hot. But what the air conditioner does, is it makes the cold even colder, and it makes the hot even hotter.

It takes some Q and it goes in that direction. Right? It takes heat from the cold room, and puts it out

into the hot air. And you're saying, this defies the second law of thermodynamics.

You have just disproved it. You deserve a Nobel Prize. And I would say to you, you're forgetting one small fact.

This air conditioner inside here, it has some type of a compressor, some type of an engine, that's actively doing this.

It's putting in work to make this happen. And this engine right here-- I'll do it in magenta-- it's also expelling some more heat.

So let's call that Q of the engine. So if you wanted to figure out the total entropy created for the universe, it would be the entropy of the cold room plus

the change in entropy for outside-- I'll call it outside, maybe I'll call this, for the room. Right?

So you might say, OK. This change in entropy for the room, it's giving away heat-- let's see the room is roughly at a constant temperature for

that one millisecond we're looking at it. It's giving away some Q at some temperature T1. And then-- so that's a minus.

And then this the outside is gaining some heat at some temperature T2. And so you'll immediately say, hey.

This number right here is a smaller number than this one. Right? Because the denominator is higher.

So if you just look at this, this would be negative entropy, and you'd say hey, this defies the second law of thermodynamics.

No! But what you have to throw in here is another notion. You have to throw in here the notion that the outside is

also getting this heat from the engine over the outside temperature. And this term, I can guarantee you-- I'm not giving you

numbers right now-- will make this whole expression positive. This term will turn the total net entropy to the universe to

be positive. Now let's think a little bit how about what entropy is and what entropy isn't in terms of words.

So when you take an intro chemistry class, the teacher often says, entropy equals disorder. Which is not incorrect.

It is disorder, but you have to be very careful what we mean by disorder. Because the very next example that's often given is that

they'll say, look. A clean room-- let's say your bedroom is clean, and then it becomes dirty.

And they'll say, look. The universe became more disordered. The dirty room has more disorder than the clean room.

And this is not a case of entropy increase. So this is not a good example. Why is that?

Because clean and dirty are just states of the room. Remember, entropy is a macro state variable. It's something you use to describe a system where you're

not in the mood to sit there and tell me what exactly every particle is doing. And this is a macro variable that actually tells me how

much time would it take for me to tell you what every particle is doing. It actually tells you how many states there are, or how much

information I would have to give you to tell you the exact state. Now, when you have a clean room and a dirty room, these

are two different states of the same room. If the room has the same temperature, and it has the same number of molecules in it and everything, then they have

the same entropy. So clean to dirty, it's not more entropy. Now, for example, I could have a dirty, cold room.

And let's say I were to go into that room and, you know, I work really hard to clean it up. And by doing so, I add a lot of heat to the system, and my

sweat molecules drop all over the place, and so there's just more stuff in that room, and it's all warmed up to me-- so to a hot, clean room with sweat in it-- so it's got more

stuff in here that can be configured in more ways, and because it's hot, every molecule in the room can take on more states, right?

Because the average kinetic energy is up, so they can kind of explore the spaces of how many kinetic energies it can have. There's more potential energies that each molecule

can take on. This is actually an increase in entropy. From a dirty, cold room to a hot, clean room.

And this actually goes well with what we know. I mean, when I go into room and I start cleaning it, I am in putting heat into the room.

And the universe is becoming more-- I guess we could say it's the entropy is increasing. So where does the term disorder apply?

Well, let's take a situation where I take a ball. I take a ball, and it falls to the ground. And then it hits the ground.

And there should have been a question that you've been asking all the time, since the first law of thermodynamics. So the ball hits the ground, right?

It got thrown up, it had some potential energy at the top, then that all gets turned into kinetic energy and it hits the ground, and then it stops.

And so your obvious question is, what happened to all that energy, right? Law of conservation of energy.

Where did all of it go? It had all that kinetic energy right before it hit the ground, then it stopped.

Right? It seems like it disappeared. But it didn't disappear.

So when the ball was falling, it had a bunch of-- you know, everything had a little bit of heat. But let's say the ground was reasonably ordered.

The ground molecules were vibrating with some kinetic energy and potential energies. And then our ball molecules were also

vibrating a little bit. But most of their motion was downwards, right? Most of the ball molecules' motion was downwards.

Now, when it hits the ground, what happens-- let me show you the interface of the ball. So the ball molecules at the front of the ball are going to

look like that. And there's a bunch of them. It's a solid.

It will maybe be some type of lattice. And then it hits the ground. And when it hits the ground-- so the ground is another solid

like that-- All right, we're looking at the microstate. What's going to happen? These guys are going to rub up against these guys, and

they're going to transfer their-- what was downward kinetic energy, and a very ordered downward kinetic energy-- they're going to transfer it to

these ground particles. And they're going to bump into the ground particles. And so when this guy bumps into that guy, he might start

moving in that direction. This guy will start oscillating in that direction, and go back and forth like that.

That guy might bounce off of this guy, and go in that direction, and bump into that guy, and go into that direction.

And then, because that guy bumped here, this guy bumps here, and because this guy bumps here, this guy bumps over there.

And so what you have is, what was relatively ordered motion, especially from the ball's point of view, when it starts rubbing up against these molecules of the ground, it

starts making the kinetic energy, or their movement, go in all sorts of random directions. Right?

This guy's going to make this guy go like that, and that guy go like that. And so when the movement is no longer ordered, if I have a

lot of molecules-- let me do it in a different color-- if I have a lot of molecules, and they're all moving in the exact same direction, then my micro state looks like my

macro state. The whole thing moves in that direction. Now, if I have a bunch of molecules, and they're all

moving in random directions, my ball as a whole will be stationary. I could have the exact same amount of kinetic energy at

the molecular level, but they're all going to be bouncing into each other. And in this case, we described the kinetic energy as internal

energy, or we describe it as temperature, where temperature is the average kinetic energy. So in this case, when we talk about, the world is becoming

more disordered, you think about the order of maybe the velocities or the energies of the molecules. Before they were reasonably ordered, the molecules-- they

might have been vibrating a little bit, but they were mainly going down in the ball. But when they bump into the ground, all of a sudden they

start vibrating in random directions a little bit more. And they make the ground vibrate in more random directions.

So it makes-- at the microstate-- everything became just that much more disordered. Now there's an interesting question here.

There is some probability you might think-- Look, this ball came down and hit the ground. Why doesn't the ball just-- isn't there some probability

that if I have a ground, that these molecules just rearrange themselves in just the right way to just hit these ball molecules in just the right way?

There's some probability, just from the random movement, that at get some second, all the ground molecules just hit the ball molecules just right to send the ball back up.

And the answer is yes. There's actually some infinitesimally small chance that that happens.

That you could have a ball that's sitting on the ground-- and this is interesting-- could have a ball that's sitting on the ground, and while you're looking, you'll

probably have to wait a few gazillion years for it to happen, if it happens at all-- it could just randomly pop up. And there's some random, very small chance that these

molecules just randomly vibrate in just the right way to be ordered for a second, and then the ball will pop up. But the probability of this happening, relative to

everything else, is essentially 0. So when people talk about order and disorder, the disorder is increasing, because now these molecules

are going in more random directions, and they can take on more potential states. And we saw that here.

And you know, on some level, entropy seems something kind of magical, but on some level, it seems relatively common sense.

In that video-- I think was the last video-- I had a case where I had a bunch of molecules, and then I had this extra space here, and then I removed the wall.

And we saw that these molecules will-- we know, there's always some modules that are bouncing off this wall before, because we probably had some pressure

associated with it. And then as soon as we remove that wall, the molecule that would have bounced there just keeps going.

There's nothing to stop it from there. In that direction, there's a lot of stuff. It could bump into other molecules, and it could

bumping into these walls. But in this direction, the odds of it bumping into everything is, especially for these leading molecules, is

essentially 0. So it's going to expand to fill the container. So that's kind of common sense.

But the neat thing is that the second law of thermodynamics, as we saw in that video, also says that this will happen. That the molecules will all expand to fill the container.

And that the odds of this happening are very low. That they all come back and go into a ordered state. Now there is some chance, just from the random movements once

they fill, that they all just happen to come back here. But it's a very, very small probability. And even more-- and I want to make this very clear-- S is a

macro state. We never talk about the entropy for an individual molecule.

If we know what an individual molecule is doing, we shouldn't be worrying about entropy. We should be worrying about the system as a whole.

So even if we're looking at the system, if we're not looking directly at the molecules, we won't even know that this actually happened.

All we can do is look at the statistical properties of the molecules. How many molecules they are, what their temperature is, all

their macro dynamics, the pressure, and say, you know what? A box that has these molecules has more state than a smaller

box, than the box when we had the wall there. Even if, by chance, all of the molecules happened to be collecting over there, we wouldn't know that that

happened, because we're not looking at the micro states. And that's a really important thing to consider. When someone says that a dirty room has a higher entropy than

a clean room, they're looking at the micro states. And entropy essentially is a macro state variable. You could just say that a room has a

certain amount of entropy. So entropy is associated with the room, and it's only useful when you really don't know exactly what's

going on in the room. You just have a general sense of how much stuff there is in the room, what's the temperature of the room,

what's the pressure in the room. Just the general macro properties. And then entropy will essentially tell us how many

possible micro states that macro system can actually have. Or how much information-- and there's a notion of information entropy-- how much information

would I have to give you to tell you what the exact micro state is of a system at that point in time. Well anyway.

Hopefully you found this discussion a little bit useful, and it clears up some misconceptions about entropy, and gives you a little bit more intuition about what it

actually is. See you in the next video.