So I mentioned in an earlier lecture, if I were to rearrange the van der Waals

equation. In order to express it with molar volume

taken to positive powers instead of appearing in denominators.

This would be the correct expression. It's v bar cubed minus quantity b plus rt

over p times v bar squared. Plus a over pv bar minus a constant, ab

divided by p equals 0. So that is a cubic equation.

And we know what cubic equations look like, they, well, they look kind of like

this actually. And it can rise, dip and rise again.

And under certain circumstances, maybe there is a point of inflection.

So, it, it's understandable why a cubic equation might be capable of reproducing

this physical behavior. And it turns out, I'm not going to show

you the equations. But the Redlich-Kwong and the

Peng-Robinson can similarly be rewritten as cubic equations in the molar volume.

And they show similar behavior in terms of being able to match pv isotherms.

So, let me take a little bit closer look at the process and remind you of what

happened in a real pv diagram. In a real PV diagram, if I start at a

lower pressure, and a large molar volume. As I increase pressure, the molar volume

drops and drops as I'm compressing. And then I would reach some point at

which in principle, the pressure remained constant while the volume continued to

drop. And then finally, I would have

transformed gas, at this point, to liquid, at this point and I would rise

substantially in pressure. What we see with the Cubic equation of

state is, so called van der Waals loops. These are unrealistic, why are they

unrealistic? Well, think about what that slope of that

line there in the isotherm means. The slope would be the change in

pressure, divided by the change in molar volume.

And in this case, it is positive, right? It's actually saying that as I increase

the pressure, I increase the molar volume.

That's what it takes to have a positive slope.

And that doesn't make any sense at all, as I increase the pressure on something,

I ought to decrease its molar volume. So that's a flaw in the equation of

state. I'll also note, I,I called out these

points A and D and said, here's where we would see the transformation from gas to

liquid. And one might ask, why there, I could

gave drawn this line down here or I could have drawn it up here?

So you can make thermodynamic arguments that I don't actually want to detour off

to derive. But something called the Maxwell equal

area construction. If you place the line where the area

above the line is equal to below the line, that's a pretty good estimate of

where the coexistence curve will be passing through.

Now, one of the points I want to make about this cubic equation of state, is

that because it's a cubic equation. At least at a temperature like this,

there are three roots to this polynomial cubic equation.

And they're real roots. But as the temperature goes closer and

closer to the critical temperature. The positions of the roots, so if I look

at root zero would be for instance here, and here, and here.

Right, they cross an axis, that cubic equation crosses an axis.

But if I get to the point where instead of going up, down, I go to a point of

inflection. That's equivalent to all of the roots

converging on a single triply degenerate root.

And that happens at the critical temperature.

Which is to say that the cubic equation at that point is molar volume minus the

critical molar volume. So that's the molar volume at the

critical temperature, cubed is equal to zero.

Right, it's a triply degenerate root. V is Vc, V bar is V bar c that is.

So, let me just expand out that cubic equation.

That very simple one. V bar minus v bar c cubed.

So all I did was cube that expression. And so I'll get v bar cube minus 3 v bar

c, v bar squared plus and so on. And the nice thing about that is let me

just compare here. I've got v cubed compares to v cubed.

Here's a term in v bar squared. So, I can relate 3 times the critical

molar volume to b plus r t over p. And I can relate 3 v bar c squared to a

over p. And, evidently, v bar c cubed is ab over

p. So, if I have this relationship between

the two equations. And I do the algebra required to actually

solve for the critical volume, the critical pressure, and the critical

temperature. I get these expressions.

So I'll let you do the algebra for yourself if you like to.

There's a whole lot of, you know, pushing terms around.

But the bottom line is the critical volume, critical molar volume, is 3 times

B. The critical pressure is a over 27 b

squared. And the critical temperature is 8a over

27 times b times the universal gas constant.