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Welcome back.

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In the last video, I told you
that pressure times volume is

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a constant.

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That if you increase the
pressure-- or if you increase

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the volume, you're going to
decrease the pressure.

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Hopefully, you got an
intuitive sense why.

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Or likewise, if you squeezed the
balloon, or the box, and

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there are no openings there,
then the pressure within the

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box would increase.

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With that said, let's see if we
can do a couple of fairly

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typical problems that
you'll see.

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So let's say that I have
a box, or a balloon, or

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something, and it has a volume,
and so let me call

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this the initial volume.

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My initial volume is 50 cubic
meters, and my initial

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pressure is 500 pascals.

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Just so you remember,
what's a pascal?

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That's 500 newtons
per meter cubed.

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I take that box, or balloon, or
whatever, and I compress it

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down to 20 meters cubed.

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So I compress it, so I squeeze
it-- that was the first

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example that I gave last time.

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It was the same container,
and I squeeze it down

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to 20 meters cubed.

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What's going to be
the new pressure?

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You should immediately have an
intuition-- what happens when

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you squeeze a balloon?

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It becomes harder to do it.

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What's going to be
the new pressure?

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It's definitely going to be
higher-- when you decrease the

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volume, the pressure increases
are inversely related.

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The pressure's going to go up,
and let's see if we can

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calculate it.

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We know that P1 times v1 is
equal to some constant, and

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since we have no aggregate
change in energy-- I'm just

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telling you that the box is
squeezed, I'm not telling you

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whether it did any work, or
anything like that-- the same

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constant is going to be equal to
the new pressure times the

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new volume, which is equal
to P2 times V2.

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You could just have the general
relationship: P1 times

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V1 is equal to P2 times V2,
assuming that no work was

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done, and there was no exchange
of energy from

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outside of the system.

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In most of these cases, when you
see this on an exam, that

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is the case.

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The old pressure was 500 pascals
times 50 meters cubed.

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One thing to keep in mind,
because this equivalence is

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not equal, and we're not saying
it has to equal some

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necessary absolute number--
for example, we don't know

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exactly what this K is, although
we could figure it

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out right now-- as long as
you're using one unit for

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pressure on this side, and one
unit for volume on this side,

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you just have to use
the same units.

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We could have done this same
exact problem the exact same

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way, if instead of meters cubed,
they said liters, as

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long as we had liters here.

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You just have to make sure
you're using the same units on

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both sides.

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In this case, we have 500
pascals as the pressure, and

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the volume is 50 meters cubed.

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That's going to be equal to the
new pressure, P2, times

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the new volume, 20
meters cubed.

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Let's see what we can do: we can
divide both sides by 10,

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so we can take the 10 out of
there, and we could divide

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both sides by 2, so that
becomes a 250.

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We we get 250 times 5 is equal
to P2, and so P2 is equal to

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1250 pascals, and if we kept
with the units, you

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would have seen that.

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When I decreased the volume
by roughly 60%, I have the

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pressure actually increased by
2 1/2, so that gels with what

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we talked about before.

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Let's add another variable into
this mix-- let's talk

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about temperature.

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Like pressure, volume, work, and
a lot of concepts that we

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talk about in physics,
temperature is something that

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you probably are at least
reasonably familiar with.

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How do you view temperature?

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A high temperature means
something is hot, and a low

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temperature means something is
cold, and I think that also

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gives you intuition that a
higher temperature object has

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more energy.

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The sun has more energy than an
ice cube-- I think that's

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fair enough.

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I think you also have the sense
that-- what would have

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more energy?

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A 100 degree cup of tea, or a
100 degree barrel of tea.

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I want to make them equivalent
in terms of

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what they're holding.

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I think you have a sense.

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Even though they're the same
temperature, they're both

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pretty warm-- let's say this
is 100 degrees Celsius, so

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they're both boiling-- that the
barrel, because there's

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more of it, is going to
have more energy.

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It's equally hot, and there's
just more molecules there.

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That's what temperature is.

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Temperature, in general, is a
measure roughly equal to some

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constant times the kinetic
energy-- the average kinetic

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energy-- per molecule.

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So the average kinetic energy
of the system divided by the

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total number of molecules we
have. Another way we could

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talk about is, temperature is

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essentially energy per molecule.

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So something that has a lot of
molecules, where N is the

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number of molecules.

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Another way we could view this
is that the kinetic energy of

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the system is going to be
equal to the number of

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molecules times the
temperature.

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This is just a constant-- times
1 over K, but we don't

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even know what this is, so we
could say that's still a

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constant-- so the kinetic energy
of the system is going

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to be equal to some constant
times the number of particles

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times temperature.

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We don't know what this
is, and we're going to

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figure this out later.

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This is another interesting
concept.

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We said that pressure times
volume is proportional to the

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kinetic energy of the system--
the aggregate, if you take all

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of the molecules and combine
their kinetic energies.

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These aren't the same K's-- I
could put another constant

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here and call that K1.

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And we also know that the
kinetic energy of the system

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is equal to some other constant
times the number of

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molecules I have times
the temperature.

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If you think about it, you could
also say that this is

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proportional to this, and this
is proportional to this.

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You could say that pressure
times volume is proportional

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to the number-- and these are
all different proportional

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constants, and we'll figure
out this exact constant

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later-- so we could say that
pressure times volume is

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proportional to molecules we
have, times temperature.

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And we said that we can
view temperature

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as energy per molecule.

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Another way we could say is
that if this constant is

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constant, which is by
definition, and the number of

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molecules is constant-- we
have PV over temperature.

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Pressure times volume over
temperature is going to be

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equal to something times the
number of molecules, so we

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could say that's some other
constant, like k4.

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This is another interesting
thing to think about: we said

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pressure times volume is equal
to pressure times volume, and

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now we added temperature
into the mix.

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We could say P1 times V1
over T1 is equal to P2

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times V2 over T2.

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Does this make sense to you?

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What happens if I have another
box, and I have my particles

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bouncing around like always.

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I have some volume, and some
amount of pressure-- what

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happens when the temperature
goes up?

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What am I saying?

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I'm saying that the average
kinetic energy per molecule is

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going to go up, so they're going
to bounce against the

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walls more.

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If they bounce against the walls
more, the pressure's

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going to go up, assuming
volume stays flat.

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Another way you could think
about it-- let's say the

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temperature goes up, and the
pressure stays flat.

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So what did I have to do?

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I just said if the temperature
goes up, the average kinetic

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energy of each molecule--
they'll bounce more.

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In order to make them bounce
against the sides of the walls

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as often, I'd have to
increase the volume.

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If you hold pressure constant,
the only way you can do that

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is by increasing the volume
while you increase the

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temperature.

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Let's keep this in mind, and we
will use this to solve some

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pretty typical problems
in the next video.