Welcome back.
In the last video, I told you that pressure times volume is
a constant.
That if you increase the pressure-- or if you increase
the volume, you're going to decrease the pressure.
Hopefully, you got an intuitive sense why.
Or likewise, if you squeezed the balloon, or the box, and
there are no openings there, then the pressure within the
box would increase.
With that said, let's see if we can do a couple of fairly
typical problems that you'll see.
So let's say that I have a box, or a balloon, or
something, and it has a volume, and so let me call
this the initial volume.
My initial volume is 50 cubic meters, and my initial
pressure is 500 pascals.
Just so you remember, what's a pascal?
That's 500 newtons per meter cubed.
I take that box, or balloon, or whatever, and I compress it
down to 20 meters cubed.
So I compress it, so I squeeze it-- that was the first
example that I gave last time.
It was the same container, and I squeeze it down
to 20 meters cubed.
What's going to be the new pressure?
You should immediately have an intuition-- what happens when
you squeeze a balloon?
It becomes harder to do it.
What's going to be the new pressure?
It's definitely going to be higher-- when you decrease the
volume, the pressure increases are inversely related.
The pressure's going to go up, and let's see if we can
calculate it.
We know that P1 times v1 is equal to some constant, and
since we have no aggregate change in energy-- I'm just
telling you that the box is squeezed, I'm not telling you
whether it did any work, or anything like that-- the same
constant is going to be equal to the new pressure times the
new volume, which is equal to P2 times V2.
You could just have the general relationship: P1 times
V1 is equal to P2 times V2, assuming that no work was
done, and there was no exchange of energy from
outside of the system.
In most of these cases, when you see this on an exam, that
is the case.
The old pressure was 500 pascals times 50 meters cubed.
One thing to keep in mind, because this equivalence is
not equal, and we're not saying it has to equal some
necessary absolute number-- for example, we don't know
exactly what this K is, although we could figure it
out right now-- as long as you're using one unit for
pressure on this side, and one unit for volume on this side,
you just have to use the same units.
We could have done this same exact problem the exact same
way, if instead of meters cubed, they said liters, as
long as we had liters here.
You just have to make sure you're using the same units on
both sides.
In this case, we have 500 pascals as the pressure, and
the volume is 50 meters cubed.
That's going to be equal to the new pressure, P2, times
the new volume, 20 meters cubed.
Let's see what we can do: we can divide both sides by 10,
so we can take the 10 out of there, and we could divide
both sides by 2, so that becomes a 250.
We we get 250 times 5 is equal to P2, and so P2 is equal to
1250 pascals, and if we kept with the units, you
would have seen that.
When I decreased the volume by roughly 60%, I have the
pressure actually increased by 2 1/2, so that gels with what
we talked about before.
Let's add another variable into this mix-- let's talk
about temperature.
Like pressure, volume, work, and a lot of concepts that we
talk about in physics, temperature is something that
you probably are at least reasonably familiar with.
How do you view temperature?
A high temperature means something is hot, and a low
temperature means something is cold, and I think that also
gives you intuition that a higher temperature object has
more energy.
The sun has more energy than an ice cube-- I think that's
fair enough.
I think you also have the sense that-- what would have
more energy?
A 100 degree cup of tea, or a 100 degree barrel of tea.
I want to make them equivalent in terms of
what they're holding.
I think you have a sense.
Even though they're the same temperature, they're both
pretty warm-- let's say this is 100 degrees Celsius, so
they're both boiling-- that the barrel, because there's
more of it, is going to have more energy.
It's equally hot, and there's just more molecules there.
That's what temperature is.
Temperature, in general, is a measure roughly equal to some
constant times the kinetic energy-- the average kinetic
energy-- per molecule.
So the average kinetic energy of the system divided by the
total number of molecules we have. Another way we could
talk about is, temperature is
essentially energy per molecule.
So something that has a lot of molecules, where N is the
number of molecules.
Another way we could view this is that the kinetic energy of
the system is going to be equal to the number of
molecules times the temperature.
This is just a constant-- times 1 over K, but we don't
even know what this is, so we could say that's still a
constant-- so the kinetic energy of the system is going
to be equal to some constant times the number of particles
times temperature.
We don't know what this is, and we're going to
figure this out later.
This is another interesting concept.
We said that pressure times volume is proportional to the
kinetic energy of the system-- the aggregate, if you take all
of the molecules and combine their kinetic energies.
These aren't the same K's-- I could put another constant
here and call that K1.
And we also know that the kinetic energy of the system
is equal to some other constant times the number of
molecules I have times the temperature.
If you think about it, you could also say that this is
proportional to this, and this is proportional to this.
You could say that pressure times volume is proportional
to the number-- and these are all different proportional
constants, and we'll figure out this exact constant
later-- so we could say that pressure times volume is
proportional to molecules we have, times temperature.
And we said that we can view temperature
as energy per molecule.
Another way we could say is that if this constant is
constant, which is by definition, and the number of
molecules is constant-- we have PV over temperature.
Pressure times volume over temperature is going to be
equal to something times the number of molecules, so we
could say that's some other constant, like k4.
This is another interesting thing to think about: we said
pressure times volume is equal to pressure times volume, and
now we added temperature into the mix.
We could say P1 times V1 over T1 is equal to P2
times V2 over T2.
Does this make sense to you?
What happens if I have another box, and I have my particles
bouncing around like always.
I have some volume, and some amount of pressure-- what
happens when the temperature goes up?
What am I saying?
I'm saying that the average kinetic energy per molecule is
going to go up, so they're going to bounce against the
walls more.
If they bounce against the walls more, the pressure's
going to go up, assuming volume stays flat.
Another way you could think about it-- let's say the
temperature goes up, and the pressure stays flat.
So what did I have to do?
I just said if the temperature goes up, the average kinetic
energy of each molecule-- they'll bounce more.
In order to make them bounce against the sides of the walls
as often, I'd have to increase the volume.
If you hold pressure constant, the only way you can do that
is by increasing the volume while you increase the
temperature.
Let's keep this in mind, and we will use this to solve some
pretty typical problems in the next video.