All these ideas are encapsulated in the quantity that we called the heat capacity.
In other words, how much energy do I have to give to some material
to raise its temperature by one degree?
So the unit of this heat capacity are joules energy
per temperature Kelvin.
So if we look at the unit as we've written it on the right here where you can see
is that heat capacity which we're writing as a capital letter C
asks us how much heat is added and also how much the temperature changes
when we add that heat.
We call the heat energy added a lower case q,
so we'll be using that letter a few times as we go.
So q has a unit of energy or joules,
and we're going to use the temperature change
which is the delta for change again and then T for temperature,
so we have the temperature change or delta T in the denominator.
As we discuss though we could also ask ourselves about the mass,
how much material is there?
Like we saw the metal bench in the same amount of volume
had a lot more mass that was there.
So we could instead define a different quantity
which we called the specific heat or the specific heat capacity
and this asks not only how much energy is required per unit temperature,
but how much energy is required per unit temperature per a given amount of mass.
So we also take into account how much mass is in our material
when we're talking about the specific heat capacity.
Possibly confusingly, we'll be using the lower case letter c
but this is the convention for the specific heat capacity
and the units are only slightly different from the heat capacity
because instead we now have the heat energy added
divided by the changing temperature times the mass
and so we also have units in mass
and the denominator for our specific heat capacity.
So notice very importantly, that is a very easy way to convert
between the heat capacity and the specific heat capacity
which is simply to multiply the specific heat capacity
the lower case c by the mass in order to get the heat capacity.
At a constant volume, it turns out that the heat capacity
will be different than the heat capacity at the constant pressure.
In other words, as we're adding energy to a gas for example,
we could pick some physical quantity to keep constant as we're adding the energy
and as we're adding the energy to increase the temperature of that system.
So again, one thing we could try to keep constant would be the volume of the gas
by keeping it at a particular amount of space,
whereas we could also do the opposite and keep the pressure constant
by putting some heavy piston maybe on top of the gas,
but allowing the gas to lift or lower that piston.
Which can change the volume but will keep the gas at a constant pressure,
because again, the gas will have to maintain a particular pressure
to hold up the weight of the piston.
The amount of heat that we add to the these two situations,
one where the volume is constant and the other one where the pressure is constant
will cause the temperature to respond differently
even if it's the same substance, the same gas.
And the reason for this, is that a constant volume system can do no work.
This is a very important idea.
So let's dig into it a little bit.
What I mean when I say a constant volume system cannot do any work,
goes back to our definition of what work is.
For something to do work as we saw it has to apply a force across a distance.
So if you look at the top piston here on the top right.
It is held at a constant volume while the gas is exerting forces
on the size of the container that it's in,
so it is applying forces and we saw that the pressure is the force per unit area.
So it has some pressure which mean just heating the container's walls.
However, because we're keeping the volume constant,
we're not allowing the piston to move, it can now apply a force over a distance.
It can't do any work,
which means it can't give up any of its energy trying to do this work.
It will retain all of its energy that we give it.
On the other hand, in a constant pressure system that we have on the bottom here,
as we add heat to our system its pressure can respond
by pushing the piston upwards changing the volume.
In this case, not only as the gas applying some sort of force
in the walls of the container,
it also moving a piston by some distance,
and so we have some work being done by the gas in the system.
When that happens, the gas again is doing work
giving up some of its energy to the piston by lifting it up in the air.
So for this reason that a constant volume system can do no work,
the heat capacity for a constant pressure system
will be greater than the heat capacity of a constant volume system,
that simply means that when you add more and more energy
to the constant pressure system.
Its temperature won't respond quite sensitively
because some of that energy will go into doing work on the piston
rather than into raising its own temperature.
So it's splitting the energy that you give it.
The heat energy, partly into doing work on the piston
and partly into raising the temperature,
which we would say increases its heat capacity,
because you can add more heat for a given change in temperature.
Again, because it's allocating that heat energy in a few different ways.
It turns out that we can actually measure the difference etween the specific heat capacity
the lower case c for a constant volume system and the constant pressure system.
We can derive this using our idealize, the model that we created
as we discussed so that for an ideal gas, as we have written here,
the heat, the specific rather heat capacity difference between the pressure
and the constant pressure and constant volume system is exactly the universal gas constant.