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# Entropy intuition

Video transcript

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 T 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.