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I've talked a lot about how our
change in internal energy

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of a system can be due to some
heat being added to the

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system, or some work being added
to the system, or being

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done to the system.

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And I'm going to write it again
the other way, just

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because you see it both ways.

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You could say that the change
in internal energy could be

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the heat added to the
system minus the

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work done by the system.

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So there's two questions
that might naturally

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spring up in your head.

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One is, how is heat added to or
taken away from a system?

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And how is work done, or done
by, or done to a system?

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The heat, I think, is
fairly intuitive.

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If I have a-- and we'll be a
little bit more precise in the

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future of this, but I just want
to give you the sense of

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what we're talking about-- if I
have some system here, some

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particles in some type
of a canister.

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And it's at temperature, I don't
know, let's say it's a

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temperature T1.

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I'll even give it a-- say
it's at 300 kelvin.

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If I want to add heat to this
system, what I can do is I can

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place another system right
next to it, maybe

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right next to it.

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Who knows what size it is.

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And it's got some
particles there.

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But its temperature is much,
much, much higher.

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So this system's temperature,
I'll say temp T2 is equal to,

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I don't know, let's say
it's 1000 kelvin.

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I'm just making up numbers.

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So what's going to happen in
this situation, you're going

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to have heat transferred from
this second system to the

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first system.

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So you're going to have heat
going into the system.

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Now, heat and, work and even
internal energy, this goes

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back to our conversation
of macrostates versus

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microstates.

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Heat is changing the macrostate
of our systems.

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This system is going to
lose temperature.

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This system's going to
gain temperature.

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But we know what's happening
on a micro level.

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These molecules are going
to lose kinetic energy.

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These molecules are going
to gain kinetic energy.

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How is that actually
happening?

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Well, we assume that there's
some type of a container here.

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Maybe it's a solid wall.

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These molecules are going to
bump into that wall, and are

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going to make the particles in
that wall vibrate, and then

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they're going to make the
particles in the green

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container's walls vibrate.

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And so when the green
container's molecules touch

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the wall, they're going to
bounce off with even more

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kinetic energy, with even more
velocity, because of that

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vibration in the wall will push
them back even further.

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So that's essentially how you
get this transfer of kinetic

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energy, or this transfer
of heat.

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I think that's fairly
intuitive.

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If we put this next to a cooler,
a system with lower

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temperature, we would
lose kinetic energy,

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or would lose heat.

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And there's other ways
that we can do it.

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We could compress the-- well,
I don't want to talk about

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that just now, because that'll
be touching on work.

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So, how can we add or subtract
work to a system?

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And this one's a little
bit more interesting.

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Let's go back to our
piston example.

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Let me just draw some
lines here.

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So I have my container.

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There you go.

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It's got a little movable
ceiling to it.

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That's my piston.

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And go back to the example.

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Because what we're going to be
dealing with-- especially once

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I go into the pressure volume
diagram, the PV diagram that

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I'm about to go into--
we want to deal with

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quasi-static processes.

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Processes that are always close
enough to equilibrium

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that we feel OK talking
about macrostates like

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pressure and volume.

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Remember, that if we just did
something crazy and the whole

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system is in flux,
those macrostates

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aren't defined anymore.

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So we want to do a quasi-static
process.

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So I'll have pebbles instead
of one big rock.

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I'll draw the pebbles a little
bigger this time.

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And I have some pressure.

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So that's my piston and
it's being kept

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down by these rocks.

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It's being kept up by the
pressure of the gas.

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The gas is bumping into
this ceiling.

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It's bumping into everything.

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The pressure at every point in
the container is the same.

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It's at equilibrium.

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Now, what happens in that
example where I removed one

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rock from that?

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So let me copy and paste that.

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So if I remove one rock from
this thing right here.

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Copy and paste.

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So that's the same thing.

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Now let me remove a rock.

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I'll remove this top
one, was removed.

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What's going to happen?

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Well I now have less weight
pushing down on the piston,

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and I have a certain amount
of pressure pushing up.

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The system, it'll very
temporarily go out of

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equilibrium, but it'll be a very
small difference in how

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much we're pressing down on it,
so hopefully it won't be a

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huge change in our
equilibrium.

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We'll stay pretty close to it.

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But we know from the previous
example, instead of this thing

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flying up, it's going to
shift up a little bit.

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This is just going to shift
up a little bit.

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Right when we do it it's going
to be like that, right there.

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And let me fill in that part
with black, because it's not

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like the space disappeared.

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So let me fill that
in right there.

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So our little piston will move
up a very small amount.

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And what I claim is, when this
happened, when I removed this

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little pebble from here, the
system did some work.

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And let's just think
about that.

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So work, according to the
definitions that you learned

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in first-year physics, and when
you're using classical or

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dealing with classical
mechanics, you learn that work

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is equal to force
times distance.

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So if I'm claiming that when
this piston moved up a little

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bit, when I removed that pebble,
I'm claiming that this

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system here did some work.

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So I'm claiming that it applied
a force to this

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piston, and it applied
that force to the

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piston for some distance.

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So let's figure out what that
is, and if we can somehow

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relate it to other macro
properties that we know

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reasonably well.

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Well we know the pressure
and the volume, right?

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We know the pressure that's
being exerted on the piston,

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at least at this
point in time.

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And what's pressure?

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Pressure is equal to
force per area.

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Remember, this piston, you're
just seeing it from the side,

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but it's a kind of a flat plate
or a flat ceiling on top

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of this thing.

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And at what distance
did it move it?

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You know I could blow
it up a little bit.

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It moved it some-- I didn't draw
it too big here-- some x,

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some distance x.

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So this change, it moved it up
some distance x there, right?

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So what is the force that
it pushed it up?

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Well, the force, we know
its pressure, the

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pressure's force per area.

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So if we want to know the force,
we have to multiply

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pressure times area.

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If we multiply both sides of
this times area, we get force.

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So we're essentially saying
the area of this little

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ceiling to this container right
there, you know, it

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could be, I could draw with some
depth, but I think you

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know what I'm talking about.

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It has some area.

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It's probably the same area as
the base of the container.

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So we could say that the force
being applied by our system--

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let me do it in a new color--
the force is equal to our

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pressure of the system, times
the area of the ceiling of our

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container of the piston.

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Now that's the force.

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Now what's the distance?

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The distance is this
x over here.

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The distance is-- I'll do it
in blue-- it's this change

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right here.

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I didn't draw it too big,
but that's that x.

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Now let's see if we can
relate this somehow.

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Let me draw it a little
bit bigger.

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And I'll try to draw in
three dimensions.

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So let me draw the piston.

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What color did I do it in?

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I did it in that brown color.

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So our piston looks something--
I'll draw it as a

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elipse-- the piston
looks like that.

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And it got pushed up.

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So it got pushed up
some distance x.

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Let me see how good
I can-- whoops.

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Let me copy and paste
that same--

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So the piston gets pushed
up some distance x.

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Let me draw that.

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It got pushed up some
distance x.

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And we're claiming that our--
oh sorry, this is the force.

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Sorry, let me be clear.

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This is the force, and
this is the distance.

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So work is equal to our force,
which is our pressure times

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our area, times the distance.

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I want to be very
clear with that.

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Because when I wrote this I
said, OK, the force that we're

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applying is the pressure we're
applying, times the area of

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our cylinder.

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This is the area of our
cylinder right here.

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That's the area of our
cylinder right there.

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So if you do the pressure
times this

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area, you get the force.

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And then we moved it
some distance x.

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Now, we could rearrange this.

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We could say that the work is
equal to our pressure times

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our area, times x.

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What's this?

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What's this area, this area
right here, times x?

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Well that's going to be our
change in volume, right?

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This area times some height
is some volume.

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And that's essentially how
much our container

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has changed in volume.

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When we pushed this piston up,
the volume of our container

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has increased.

217
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You can see that, even looking
from the side.

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Our rectangle got a
little bit taller.

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When you look at it with a
little bit of depth, you see

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the rectangle also didn't
get taller.

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We have some surface area.

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Surface area times
height is volume.

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So this right here, this
term right here,

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is a change in volume.

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So we can write work now in
terms of things that we know.

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We can write work done
by our system.

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Work done is equal to pressure
times our change in volume.

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Now this has a very interesting
repercussion here.

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So we could-- actually many--
we can rewrite our internal

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energy formulas.

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So, for example, we can write
internal change and internal

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energy is now equal to heat
added to the system, plus the

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work-- let me say minus the
work done by the system.

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00:11:08,565 --> 00:11:12,450


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Well what is the work
done by the system?

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Well it's the pressure of the
system times how much the

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system expanded.

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In this case, the system is
pushing these marbles, or

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these pieces of sand up.

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It's doing work.

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If we were doing it the other
way, if we were adding the

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sand, and we were pushing down
on our little canister, we

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would be doing work
to the system.

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00:11:35,300 --> 00:11:37,460
So this is the situation where
I'm doing here, where I'm

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removing the sand and the piston
goes up, essentially

247
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the gas is pushing up on
the piston, the system

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is doing the work.

249
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So if we go back to our little
formula, that internal energy

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is heat minus the work done by
a system, so done by, then we

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can write this as, this is equal
to the heat added to the

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system minus this quantity, the
pressure of the system,

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times the change in volume.

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And it's interesting, if the
volume is increasing, then the

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system is doing work.

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And this applies-- we're going
to talk a lot more about

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engines in the future-- but
that's how engines do work.

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They have a little explosion
that goes on inside of a

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cylinder that pushes up on the
piston, and then that piston

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moves a bunch of other stuff
that eventually turns wheels.

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So the volume increases, you're
actually doing work.

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So I'm going to leave you
there in this video.

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In the next video, we're going
to relate this, this new way

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of writing our internal energy
formula, and we're going to

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relate it to the PV diagram.

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