- [Instructor] Imagine you had a monatomic ideal gas
in this cylinder here,
and there was this tightly-fitted piston above it
that prevented any gas from getting out.
What we know that the total internal energy,
for a monatomic ideal gas is just
3/2 P times V,
or 3/2 NKT,
or 3/2 little nRT.
And we know that saying U internal,
the internal energy is really just code for
the total kinetic energy of the monatomic ideal gas.
These are the same thing.
When we talk about internal energy,
we're talking about how fast are these particles moving,
in other words, what's the total kinetic energy
of all of these particles added up?
My question is, how do we go about changing
the internal energy?
Let's say we wanted to increase the total kinetic energy.
What could we do?
Well, we could say, increase the pressure
or the volume or the temperature, yeah, but, I mean,
physically, actually, in the lab, what do we do?
And there's basically two ways
to change the internal energy.
If we want to add internal energy,
i.e. get these particles moving faster,
we can heat it up so put this above a flame,
or on a hot plate,
and heat will flow into the gas,
which will cause these particles to move faster and faster.
That's one way to do it, to add heat.
The other way to do it, is to do work on the gas.
I could take this piston and push it down.
And if you push this down hard enough,
it'll squash this gas together,
and those impacts, with this piston while it's moving down
will cause them to start moving faster and faster.
That will also add internal energy to the gas.
So if we wanted to write down a formula
that told us how you could get a change
in the internal energy,
if I want to change the internal energy, delta U,
which is really just saying changing the kinetic energy.
Well, there's two ways to do it.
I could add heat.
If I added ten joules of heat.
I add ten joules to the internal energy.
But I've also gotta take into account this work being done,
and so I could do plus the work done on the gas,
and that's it.
This is actually the First Law of Thermodynamics.
It's the law of conservation of energy.
It says there's only two ways to add energy,
internal energy, to a gas.
Let me talk a little bit more about this work done though,
'cause getting the sign right's important.
If you're doing work on the gas, compressing it,
you're adding energy to the gas.
But, if you let the gas push up on the piston,
and this gas expands, pushing the piston up,
then the gas is doing work,
that's energy leaving the system.
So if the gas does work,
you have to subtract work done by the gas.
If the outside force does work on the gas,
you add that to the internal energy.
So you've gotta pay attention
to which way the energy's flowing.
Work done on gas, energy goes in.
Work done by the gas, energy goes out,
and you'd have to subtract that over here.
Let's say the gas did expand.
Let's say the gas in here was under so much pressure
that the force it exerted on this piston
was enough to push that piston upward by a certain amount.
So let's say it started right here and it went up to here,
so that piston went from here to there.
No gasses escaping, 'cause this is tightly fitting,
but the gas was able to push it up a certain distance d.
How much work was done?
You know the definition of work.
Work is defined to be the force times the distance
through which that force was applied.
So the work that the gas did, was f times d,
but we want this to be in terms of thermal quantities,
like pressure and volume and temperature.
So what could we do?
We can say that this volume,
not only did the piston raise up,
but there was an extra volume generated within here.
That I'm gonna call delta v.
And I know that this delta v has got to equal
the area of the piston,
times the distance through which that piston moved.
Because this height times that area
gives me this volume right in here.
Why am I doing this?
'Cause look, I can write d as equal to delta V over A,
and I can take this, and I can substitute this formula for d
into here, and something magical happens.
I'll get Work equals f times delta V over A.
But look, f over A.
We know what f over A is, that's pressure.
So I get the work done by the gas
is the pressure times delta V.
This is an equation that I like
because it's in terms of thermodynamic quantities
that we're already dealing with.
So work done, you can figure out
by taking P times delta V.
But strictly speaking,
this is only true if this pressure remains constant, right?
If the pressure was changing,
then what am I supposed to plug in here?
The initial pressure, the final pressure?
If the pressure's staying constant,
this gives you an exact way to find the work done.
You might object and say wait, how's it possible for a gas
to expand and remain at the same pressure?
Well, you basically have to heat it up
while the gas expands.
That allows the pressure to remain constant
as the gas expands.
And now we're finally ready to talk about Heat Capacity,
so let's get rid of this.
Heat Capacity is defined to be,
imagine you had a certain amount of heat being added.
So a certain amount of heat gets added to your gas.
How much does the temperature increase?
That's what the heat capacity tells you.
So capital C is heat capacity,
and it's defined to be the amount of heat
that you've added to the gas,
divided by the amount of change in the temperature
of that gas.
And actually, something you'll hear about often
is the Molar Heat Capacity,
which is actually divided by an extra n here.
So instead of Q over delta T,
it's Q over n, the number of moles, times delta T.
Pretty simple, but think about it.
If we had a piston in here,
are we gonna allow that piston to move while we add the heat
or are we not going to allow the piston to move?
There's different ways that this can happen.
And because of that, there's different heat capacities.
If we don't allow this piston to move,
if we weld this thing shut so it can't move,
we've got Heat Capacity at constant volume,
and if we do allow this piston to move freely,
while we add the heat,
so that the pressure inside of here remains constant,
we'd have the Heat Capacity at constant pressure.
And these are similar, but different, and they're related.
We can figure them out.
So let's clear this away, let's get a nice, here we go.
Two pistons inside of cylinders.
We put a piston in here, but I'm gonna weld this one shut.
This one can't move.
We'll have another one over here, it can move freely.
So over on this side,
we'll have the definition of Heat Capacity,
regular Heat Capacity is the amount of heat you add,
divided by the change in temperature that you get.
So on this side, we're adding heat, let's say.
Heat goes in, but the piston does not move.
And so the gas in here is stuck, it can't move.
No work can be done.
Since this piston can't move,
external forces can't do work on the gas,
and the gas can't do work and allow energy to leave.
Q is the only thing adding energy into the system.
Or in other words,
we've got Heat Capacity at constant volume, is gonna equal,
well remember the First Law of Thermodynamics,
said that delta U, or the only way to add internal energy,
or take it away,
is that you can add or subtract heat,
and you can do work on the gas.
In this case, Q, if I subtract W from both sides,
I'd get delta U minus W over delta T.
But since we're not allowing this piston to move,
the work done has got to be zero.
So there's no work done at all,
so the Heat Capacity at constant volume
is gonna be delta U over delta T.
What's delta U?
Let's just assume this is a monatomic ideal gas.
If it's monatomic, we've got a formula for this.
Delta U is just 3/2, P times V over delta T.
That's not the only way I can write it.
Remember, I can also write it as 3/2, NK delta T,
over delta T, and something magical happens.
Check it out, the delta Ts go away,
and you get that this is a constant.
That the Heat Capacity for any monatomic ideal gas
is just gonna be 3/2,
capital NK Bol so it's constant.
N is the total number of molecules,
or you could have re-wrote this as little nR delta T.
The Ts would still have canceled.
And you would've got 3/2, little n,
the number of moles, times R, the gas constant.
So the Heat Capacity at constant volume
for any monatomic ideal gas, is just 3/2 nR.
And if you wanted the Molar Heat Capacity,
remember that's just divide by an extra mole here,
so everything gets divided by moles, everywhere,
divided by moles, that just cancels this out,
and the Molar Heat Capacity at constant volume,
is just 3/2 R.
So that's Heat Capacity at constant volume.
What about Heat Capacity at constant pressure?
Now we're gonna look at this side.
Again, we're gonna allow this gas
to have heat enter the cylinder,
but we're gonna allow this piston to move up
while it does that,
so that the pressure inside of here remains constant.
And this is gonna be the heat capacity at constant pressure.
Well, again, we're gonna get that it's Q over delta T,
and just like the first law said,
Q has to equal delta U minus W.
So we get delta U minus W, the work done, over delta T.
This time W's not zero.
What's W gonna be?
Remember, W is P times delta V.
So this is a way we can find the work done by the gas,
P times delta V.
So this is gonna equal Delta U, we know what that is.
If this is again a monatomic ideal gas,
this is gonna equal 3/2, nR delta T plus,
this is P times delta V but we have to be careful.
In this formula, this work is referring to
work done on the gas.
But in this case, work is being done by the gas,
so I need another negative.
Technically the work done on the gas,
would be a negative amount of this,
since energy is leaving the system.
So that negative cancels this negative,
and I get plus P times delta V.
All of that over delta T.
Whew.
So, what do we get?
3/2, nR delta T, plus,
I wanna re-write P times delta V,
but I know how to do that.
The ideal gas law says PV equals nRT.
Well if that's true, then P times delta V
is gonna equal nR delta T.
So I can re-write this as nR delta T, divided by delta T.
Almost there, all of the delta Ts go away,
and look what I'm left with.
I'm left with C.
Heat Capacity at constant pressure
is gonna be equal to 3/2 nR, plus nR.
That's just 5/2 nR.
And, if I wanted the Molar Heat Capacity again,
I could divide everything around here by little n,
and that would just give me the Molar Heat Capacity
constant pressure would be 5/2 R.
And notice they're almost the same.
The Heat Capacity at constant volume is 3/2 nR,
and the Heat Capacity at constant pressure is 5/2 nR.
They just differ by nR.
So the difference between the Heat Capacity
at constant volume, which is 3/2 nR,
and the Heat Capacity at constant pressure,
which is 5/2 nR,
is just Cp-Cv,
which is nR.
Just nR.
And if you wanted to take the difference
between the Molar Heat Capacities,
at constant volume and pressure, it would just be R.
The difference would just be R
because everything would get divided by the number of moles.
So there's a relationship, an important relationship,
that tells you the difference
between the Heat Capacity at constant pressure,
and the Heat Capacity at constant volume.