And yet another equation, is the Peng-Robinson equation of state.

Once more, a somewhat complex relationship more complex perhaps than

the Van Der Waals, and the relevant parameters are not called A and B

anymore, they're called alpha and beta. But in any case, they serve a similar

function. They are designed to say something about

attractiveness or repulsiveness of the molecules in the gas phase.

And I've called these cubic equations of state, in the title of the slide, because

if we were in fact, to expand these out, take these V bars out of the denominator,

and I'll show you examples of that in not too long, we would end up with an

equation that is cubic in V bar. And that has key implications, and let's

explore that momentarily. But first let me just actually show you

some of this behavior. And so on this graph I've go pressure on

the left hand side on the Y-axis, and density on the X-axis.

So we haven't seen density before, but let me just note for you, density is just

the inverse of molar volume. And so that's sort of obvious if you

think about the units. Density is mols per liter, for example.

And that, and molar volume is liters per mol.

So they're just related to one another as the inverse.

And you can write equations either way, if you like to.

and this is just something to keep in mind, because we may use either of these

units in the future. A liter is a cubic decimeter, so

sometimes thermodynamics prefer decimeters to liters.

In any case these are data for ethane as a gas at 400 k, and shown here are the

experimental data for the change in density, as a function in the change in

pressure. And so what you see is that, as the

density is increasing, that is as you're compressing the gas, you're putting more

mass into a smaller volume. The pressure is going up, you must

squeeze on the gas to force it into that volume.

So the experiment is the solid line here, and that does not mean solid phase

ethane, it's just the solid line on the graph.

you see the, the Van Der Waals equation of state, does well up to about a density

of maybe, 7 it looks like. And then it starts to predict that much

higher pressures would be required, to keep compressing that are observed.

On the other hand, Redlich-Kwong and Peng-Robinson are doing reasonably well.

And so those somewhat more complex equations of state, have improved

performance at very very high pressures. So let's actually cool our ethane down a

little bit. Let's take it does to 305.33 degrees

kelvin. At that temperature, we observe something

happen, as we keep pressing on a piston. So if you imagine that you've got a

volume in here of gas, and I'm decreasing that volume by pressing on a piston.

At a certain point, I would observe that the pressure I'm applying, I don't have

to increase pressure but the piston keeps going down, and if I look inside my

container I would discover, that's because my gas is liquifying.

I'm seeing a liquid phase appear at the bottom of the container.

And that will continue, until the piston finally touches the top of the liquid,

and now suddenly I'm going to have to increase the pressure dramatically to

compress it any further. So, that's what's really being shown

here, the experiment is again the solid line.

I've increased pressure up, up, up. And then suddenly I hit this flat part,

where keeping the pressure constant, the density just keeps getting larger and

larger, and that's the average density. because the density of a liquid, is much

larger than the density of a typical gas. And so, it'll transform itself from gas,

all the way over to liquid, with its normal liquid density, at that pressure

and temperature. And then if I want to keep compressing a

liquid, that's much harder than compressing a gas, and the pressure

shoots up. So we say that the liquid and the vapor

are in equilibrium over this range. They're both present, and until I go

beyond a certain pressure, they will both be present.