At constant pressure, on the other hand, we will define a constant pressure heat

capacity as, the change in enthalpy with respect to the change in temperature.

Because at constant pressure the heat is equal to the enthalpy.

Alright, so the amount of heat Required to change the temperature by one degree,

that would make the denominator one degree.

That is the heat capacity. That's the definition.

So partial h, partial t at constant pressure.

So with those definitions in mind, let me give you a chance to think about what

their relationships might be and then we'll come back and look at them more

closely. Alright, so the question there was, which

of the two heat capacities, constant volume or constant pressure would be

larger. Lemme take a specific example, now that

you've had a chance to think about the answer, and hopefully answer correctly.

Then we're going to get to here. let me do an ideal gas, and so recall for

an ideal gas definition of enthalpy is h equals u plus pv.

And, so, for an ideal gas, pv is nrt. If I differentiate both sides with

respect to temperature, I'll get dh dt is equal to du dt plus nr.

Now, for an ideal gas, you and hence, h, depend only on temperature, not on

pressure or volume. so, I can write this more generally, this

exact differential is also equal to the partial of h, with respect to t.

It's just a differetn way to write it. There are no other things h depends on,

but I'll use the partial symbol. At constant pressure, equals partial u,

partial t, at constant volume plus nr. That is, cp equals cv plus nr.

So the heat capacity at constant pressure is always going to be greater than the

heat capacity at constant volume. How much great for an ideal gas?

By a factor of n, number of moles of the gas, times r.

And I'll just remind you if, if you haven't got a feel, is that a lot, a

little? Well remember that for a monatomic ideal

gas, the molar heat capacity at constant volume was three halves r.

So that would make the heat capacity, the molar constant pressure heat capacity,

five halves r. That's, that's a 67% change, right?

r is 67% of three halves r. and that's a nontrivial difference.

So it takes considerably more heat added to raise the temperature by one degree

when working at constant pressure, compared to when working at constant

volume. So and of course the reason by the way is

that you've got to expand the gas. You're doing pv work and you've gotta put

the heat in to do that. Well the reason this is interesting, this

heat capacity at constant pressure is, it allows us to potentially determine

enthalpy experimentally. Or more accurately, perhaps enthalpy

changes. Remember, thermodynamics is usually about

enthalpy changes. But, we'll get to establishing zeros and

being able to tabulate numbers in not too long.

But let's talk about determining enthalpy then.

The difference in enthalpy at two different temperatures Is going to be

determined by integrating cp over the temperature range.

We've already seen this for internal energy, integrating cv.

Now I'm going to take cp equals partial h partial t.

So I rewrite this as dh equals cpdt. So, if I want to know h2 minus ht1, I

integrate dh from t1 to t2, that's equivalent to integrating this from t1 to

t2, and here that is, integral t1 to t2 cpdt.

Now, it's important to mention this is only true if the phase of the system

remains unchanged over that temperature range.

t1 to t2. We actually did an example not so long

ago of ice melting or water boiling. That's a phase change and it takes

additional heat which is enthalpy at constant pressure added to the system in

order to accomplish a phase change. So, at a phase change the heat capacity

becomes infinite, right? You are adding heat into the system

without changing the temperature. So the denominator if you like is, is

zero, and that's why it goes to infinity. but you can measure heat capacity over

the range of a pure phase, and then you'll eventually get to a phase change

and you'll need to measure that differently.

But it, it can be measured. And so, if you like, a way to think about

crossing a phase boundary would be if I want to know the enthalpy at a

temperature t and I'll start from zero. I'll have to assign some number to the

enthalpy at absolute zero, we can talk about how you might do that, but anyway

there is some number associated with that.

How would I do it? Well, I'll integrate from zero up to the

melting point. I'm going to assume that it's absolute

zero, it's probably a solid. I'll integrate the solid's heat capacity

up to the melting point, then, I will add the enthalpy of fusion.

And then, I'll integrate from that temperature, which is still the same

temperature, the temperature of fusion, up to the temperature of interest t, the

heat capacity of the liquid. And it'll have a distinct heat capacity

from that of the solid. So just what I said, solid from t equals

zero to t fusion, enthalpy of fusion, which is the enthalpy of the liquid,

minus the enthalpy of a solid at that fusion temperature, and then the liquid.

So let me show you what that looks like in practice.

Show you some experimental data. Benzene, so benzene and aromatic organic

molecule. found in oil and it has a melting point

of 278.7 kelvin and a boiling point of 353.2 kelvin.

And if you measure its heat capacity, temperature by temperature.

And what does that mean to measure the heat capacity?

It, it's pretty simple. You put a thermometer in the substance,

you add heat in some controlled fashion that you can quantify how much heat

you're putting in. And you measure how much you put in to

get it to go up one degree. Maybe that's how many meters of methane

did you burn in a Bunsen burner? Maybe it's how many calories of sugar did

you expend while you were turning a crank?

That would be a hard experiment to do, but in any case you can measure it.

And you can measure it degree by degree. So, it does vary, this is temperature on

this axis, this is the constant pressure heat capacity and what you see is, at

absolute zero it takes hardly any. And then, as you rise in temperature,

it's taking increasingly more energy added in order to raise the temperature.

And that's because that energy is flowing into more accessible modes.

Rotations, vibrations are beginning to pick up some of the energy, and they're

not being put into translation, which increases temperature.

That's really a gas explanation, and we're, we're sitting in a solid region,

but it's the same conceptual ideas. In any case, we rise, we rise, we rise,

we finally hit the melting point. Here the heat capacity would go infinite.

We can't measure it directly. We would have to do a, a measurement

where we just watch until all the solid melts and becomes liquid.

And we know how much heat we put in, so that's the heat of fusion.

And now we go measure the liquid and then, it boils and we measure the gas,

and this would be key experimental data. The enthalpy itself then, because these

are enthalpy changes with respect to temperature, to get the enthalpy is the

integral under these curves. And so that's what's shown over here.

As I raise the temperature, what is the additional enthalpy, that is the

additional heat that's been absorbed into the system, that's there.

I can extract it maybe to do interesting things, as a function of temperature.

So I don't have anything in there when I start, and I'm pouring it in, pouring it

in. It's going up, up, up as a solid, and

then the temperature doesn't change anymore but I pour in a whole bunch to

make it change phase. I pour in more to increase the enthalpy

of the liquid. Stops absorbing but does start turning

into a gas. Finally I've got all this enthalpy in the

gas. So I'm integrating that is for t above

the vaporization temperature. The enthalpy relative to zero would be

integral over the solid, plus the, the heat of fusion.

Integral over the liquid, plus the heat of vaporization.

And then finally integral over the temperature to whatever temperature I'm

at in the vapor. Alright, hopefully that made clear.

The, the experiments are relatively simple, and you would have a way to

record the enthalpy change relative to absolute zero.

Might not be the most convenient place to anchor your scale to.

It might be hard to do a measurement at absolute zero.