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# PV-diagrams and expansion work

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In the last video, we
saw that a system could do work by expanding. And in the situation we drew,
we had a situation where the ceiling was movable. We had this piston and we, like
in our process video, we had a bunch of pebbles. We removed a pebble, so the
pressure in our system, if we assume that it was just so small
that the pressure was constant, it pushed up on the
piston with some force. We figured out that that force,
since pressure is force per area, we just multiplied
pressure times the area of our piston, and we got the amount
of force we're applying. We apply that, and then we
multiply that times the distance that we push the piston
up, and then we get the amount of work that it
did by expansion, or the expansion work. We said, well, you know we could
have rewritten that. If you said pressure times our
area, times our distance, we could instead write that as
pressure, times the area, times the distance. And the area times
the distance is the change in volume. And so we came up with a neat
little formulation, that the work done by a system could be
written as the pressure times the change in volume. So in this case, I wrote the
internal energy formula, where it's the work done
by the system. So I did a minus, right? Because when you do work, you
are giving energy away to someone else. So in that situation,
we did a minus. And so instead of writing work,
we could say, minus the pressure, times the
change in volume. And remember this is a
quasi-static process. And we're doing it at very
small increments. We're assuming that this change
in volume is very small, and that the pressure
is roughly constant while we're doing this. And of course that's not
the case, right? If we did this, if this was a
large change in volume, or if this happened all of a sudden,
if these were really big pebbles, then our pressure
will change as we expand. So it's hard to say what
the pressure times the change in volume is. But if we assume things are
really, really being done in very, very small increments,
we could say, OK, let's say the pressure was constant over
that small increment, and then we can multiply it by the
change in volume. Now let's see how this can
relate to some of what we've done before with
the PV-diagram. And so far, all we've seen the
PV-diagram, or what I used it for, is to kind of help explain
the difference between quasi-static processes,
or to say when macrostates are defined. But let me now do something
more useful with it. And this will give you an idea,
or start giving you an idea of why people who
study thermodynamics love these so much. So before I did anything, when
my canister was just here, I had all the pebbles on it. And we were in a state
of equilibrium. I could describe all of its
macrostates, its pressure, its volume, its temperature. I could describe its internal
energy as well. So let me draw it here. So let's say I was
at this state. This was state number 1. State number 1 was
right there. And then, let's say I just
start removing pebbles. Remember, if I just remove all
the pebbles at once, the system's going to
go into flux. We wouldn't be doing a
quasi-static process, or a reversible process, which isn't
always the same thing. But for our purposes, we
wouldn't be in equilibrium the whole time. And we would have to wait
to get to equilibrium. And at some point we'd have
some pressure and volume that's down here. This is if we weren't doing it
as a quasi-static process. Now we are, what I showed in the
last video, we are doing it as a, or we're trying to get
close to a quasi-static process, because we're doing it
in small increments, with these little pebbles. And if these aren't small enough
for you, you could do it in smaller pebbles. So we're moving incrementally. So, for example, in that
last video, we maybe moved from there. We removed one pebble and
we got right there. You remove another pebble
and you go right there. You remove another pebble
and you go right there. And the benefit of doing these
quasi-static processes is you really get a path going from
one state to the next. Let's say when you remove all
but one of the pebbles, just, you know, this describes
our path. So let's say we are in state 2
and we've removed all but one. Let me draw that. So state 2 will look something
like this. I'll draw it really quick. So that's our container. That's our piston. We only have one pebble
left on top. And then of course we
have the gas now. Let me write this down. This is state 2. And let me write state 1 was
something like this. State 1, the ceiling
was lower. We had a bunch of pebbles
on top of it. And we had a smaller volume,
and so the gas was bumping into the ceiling and the walls
and the floor a lot more. I'll just draw the
same number. So we had a higher pressure. So pressure was high
and volume low. Now in state 2-- so this
is pressure high. This pressure is this axis. This is volume. So we had high pressure,
low volume. And we got to a situation after removing all but one pebble. And we're doing it slowly, so
we're always in equilibrium. So we have a path. This is after removing each of
the pebbles, so that our pressure and volume
macro states are always well defined. But in state 2, we now
have a pressure low and volume is high. The volume is high, you can just
see that, because we kept pushing the piston up slowly,
slowly, trying to maintain ourselves in equilibrium so our macrostates are always defined. And our pressure is lower just
because we could have the same number of particles, but they're
just going to bump into the walls a little bit
less, because they have a little bit more room
to move around. And that's all fair and dandy. So this describes the path of
our system as it transitioned or as it experienced this
process, which was a quasi-static process. Everything was defined
at every point. Now we said that the work done
at any given point by the system is the pressure times
the change in volume. Now, how does that
relate to here? Change in volume is
just a certain distance along this x-axis. Along, more like I should
call it the volume-axis. This is a change in volume. We started off at this volume,
and let's say when we removed one pebble we got
to this volume. Now, we want to multiply that
times our pressure. Since we did it over such a
small increment, and we're so close to equilibrium, we could
assume that our pressure's is roughly constant over
that period of time. So we could say that this
is the pressure over that period of time. And so how much work we did,
it's this pressure over here, times this volume, which is
the area of this rectangle right there. And for any of you all who've
seen my calculus videos, this should start looking a
little bit familiar. And then what about when we
could take our next pebble? Well now our pressure is
a little bit lower. This is our new pressure. Our pressure is a little
bit lower. And we multiply that times our
new change in volume-- times this change in volume-- and we
have that increment of work. Once again, this is the area
of this rectangle. And if you keep doing that, the
amount of work we do is essentially the area of all of
these rectangles as we remove each pebble. And now you might say,
especially those of you who haven't watched my calculus
videos, gee, you know, this might be getting close, but the
area of these rectangles isn't exactly the area
of this curve. It's a little inexact. There's a little error here. And what I would say is, well if
you're worried about that, what you should do is use
smaller increments of volume. And if you want to have smaller
changes in volume along each step, what
you do is you remove even smaller pebbles. And this goes back to trying
to get to that ideal quasi-static process. So if you did that of,
eventually the delta V's would get smaller and smaller and
smaller, and the rectangles would get thinner and
thinner and thinner. You'd have to do it over
more and more steps. But eventually you'll get to a
point, if you assume really small changes in our delta V. In calculus world, that
infinitely small changes, you write it as dV. So if you take a sum of all the
pressures, times the dV's you get the area under
this curve. So the way to think about it
when you're looking at this PV-diagram, if someone says,
you're going from this point to this pressure and this
volume, to this pressure and this volume. And they say, how much
work did you do? You say, oh, well I just had to
figure out the area under this curve. If you want to know the real
math behind it, if you could get your pressure as a function
of volume-- and if you haven't watched the calculus
videos you can ignore this little aside I'm
going to do here. This is this curve right here. If you could write it this way,
let's say you could write pressure as a function
of volume. When you're in algebra, you
learn a curve is, you know, y is a function of x. But here, y is the pressure
and x is volume, so its pressure is a function
of volume. So the area under this curve
is the integral of the pressure as a function of
volume, that's the height at any point, times our very
small change in volume. So times our very small
change in volume. And you take the sum from our
starting volume, so volume initial to volume final. And we'll do this in the future,
especially when we start touching on entropy. But this is a neat result. Even if you don't know the
calculus, or if this confuses you, if you've never seen
an integral before, you could ignore it. But you could look at this
intuitively and see the work I did is the area under
this curve. Now, let me ask you
one more thing. Let's say some work is being
done to the system. So we start adding some
marbles back. So let's say-- actually,
let's say we're going from this direction. Let's say we start at state 2
and we go in that direction. So direction matters. So let's say we go in that
direction right there. So I should put some arrows. And I'm overloading this
picture so much. Actually let me just do a new
picture, that's probably the best thing to do. So it's pressure, volume-- I'm
actually going to do two. Let me just do pressure,
volume. I'm going to do two
graphs here. All right. So in the first one
it's pressure, volume, pressure, volume. We started here at 1, and
we went here to 2. So our system was essentially
pushed up on the piston. And it could be a curve or a
line, I'm not going to get too particular right now, but it was
going in this direction. And so we can say that the work
done was the pressure times the increase in volume
at any moment. So the work done was the
area under this curve. Now, if we started at position
2 and we go to position 1. 2 to 1. Now what's happening? Now we're compressing. So if we're going in that
direction, you might say, oh OK, maybe the work done by
the system is still the area under the curve. Well you'd be close. Because what's happening now? We're now compressing
the system or adding the marbles back. We're putting energy
into the system. So if we do that, remember, your
work done by the system was pressure times an
increase in volume. Now it's going to be
your pressure times a decrease in volume. So when you go back in this
direction, the area is not the work done by the system, it's
the work done to the system. And maybe I'll do that in a
different color, so green for work done to the system. Now let me throw you another
little interesting idea. And this is actually
a key idea. It's good to get the
intuition here. So let me just draw a very
simple PV-diagram again. So let's say we start
at some state here. State 1. And I do something, you know,
I'm in a quasi-static process and it, you know, it's doing
something weird, and I get to state 2 here. And it's going in
this direction. So my volume is increasing. So in this situation, what is
the work done by the system? Easy enough, it's the area
under this curve. Now let's say that I keep
doing some type of quasi-static process, but it
takes a different path. I'm doing something else, other
than adding the marbles directly back. So my new path looks something
like this to get back to state 1. So these arrows are
going back. So now what is the work
done to the system? well my volume is decreasing,
so it's the area under the second curve. The area under the second
curve is the work done to the system. So if I want to know what the
net work the system did, going from state 1 to state 2, and
then going back to state 1-- remember, this is a pressure
and volume diagram-- what is it? Well the work that the system
did was this whole area under this brown curve. And then it had some work done
to it, which is the area under this magenta curve. So the net work it did is
essentially the white, the whole area, minus
this red area. So the net work it did would be
essentially just the area inside this loop. And hopefully you don't have to
know calculus to do this, although calculus you would
actually use to compute these areas. But I just want to give you that
intuition, that the area inside this closed loop is
actually the amount of work that our system has done. And what's important is the
direction that it's going. So it increased volume, then
decreased volume, so it's kind of this clockwise motion. This is the work that our system
has done, which, I don't know, to me is a pretty
interesting thing. And later we can use this notion
to come up with some other ideas behind our
state variables I'll make one little
aside here. Remember, our state variable
pressure volume, we did stuff to it then we went back
to that state. That stayed the same. And I want to say
another thing. For our purposes, when we're
dealing with ideal gases, where the internal energy is
essentially the kinetic energy of the system, if we go and do
all sorts of crazy stuff and come back, our internal
energy hasn't changed. So the internal energy is always
going to be the same at this point. So if I said, I did all of this
stuff and came back here, what is my change in
internal energy? It's 0. The change is 0. Now if I said I went from here
to here, I would have a different internal energy
and my change would be something real. But since this is a state
function, it doesn't care how I got there. If I took all these loops and
got back there, it just says, look, if I'm at this point in
the PV-diagram, my internal energy is the same thing. So if I start at this point,
and I finish again at this point, I have had no change
in internal energy. And we'll talk more about
that in the next video. But I just wanted to leave you
there and get you this intuition behind the areas
under the curves in the PV-diagram.