So, let's take a loo at another property which shows corresponding state behavior.

And in particular, let's look at the compressability factor Z, and in this

instance, you'll recall the compressability factor is defined as

pressure times molar volume, divided by, the universal gas constant times

temperature. And I want to take a moment for an aside

here. Compressibility factor is really the best

thing to call this quantity Z. Occasionally, I've been a bit colloquial

and I've referred to it as just compressibility, which is, in some sense

a little counter-intuitive. If you think about the values the

compressibility factor takes on. If it's a number greater than 1, which is

to say that the product of pressure in molar volume is larger than it would be

for an ideal gas. the way to think about that is given the

pressure, perhaps the molar volume is larger than it would be for an ideal gas,

which is to say It's a bit hard to compress the real gas compared to the

ideal gas. So, there's this unusual feature in a way

that the compressibility factor is larger when the actual compressibility, if you

want to think of that as ability to be compressed, is smaller.

But, okay, we'll ignore that paradox for a moment, and I'll try to be a little

more clear and say compressibility factor here.

But if we want to think of a universal compressibility factor within the context

of the Van der Waal's equation of state. If I replace pressure using the Van der

Waal's equation of state with its expression in terms of v bar and t.

And I work at the critical temperatures, for instance, and I substitute in all the

a's and b's with their corresponding relationships that are defined for the

Van der Waal's parameters in relation to the critical parameters, pressure

temperature and molar volume. Then, with a lot of algebra, and I won't

go through all the algebra, it's a little tedious.

But if, if you wanted to work backwards. You could actually plug in for these

reduced values. What they are by definition, and then

replace the critical values with the dependents on the Van der Waals

constants. And find that you would ultimately walk

your way back to the Van der Waals equation of state value for pressure

plugged into compressibility factor. But I'll, I'll leave that for the people

who really want to do that algebra. But, the take home message is, that the

compressibility factor becomes expressed exclusively in terms of two reduced

variables. So, here are the reduced molar volume and

the reduced temperature. And, of course, I could, continue to,

manipulate this to have it in terms of any two variables.

But what it means is, I ought to be able to plot the compressibility factor

against any two reduced variables. So, in this particular instance, I'm

going to have the reduced pressure on the x axis.

And then a series of reduced isotherms, that is constant values of reduced

temperature ranging from 2 to 1. So, 1 of course would be actually at the

critical temperature. And although the symbols are probably a

little bit small that, to be completely clear.

These different symbols filled circles, open circles, triangles, half filled, and

so on, all correspond to different gases. But when we plot them using their reduced

temperatures, and reduced pressures, they all fall on equivalent reduced isotherms.

And so if I want to know, say, the compressibility of any gas when it is at

its critical temperature, that would be tr equal 1, and at its critical pressure,

that would be Pr equal 1, which would be right around here, I can just read it

off. I don't need to know the nature of the

gas. They all ought to behave about the same.

They all ought to have a compressibility factor a little in excess of 0.2.

So, again, that's a, a, an example of corresponding conditions leading to

equivalent behavior for all gases. So, I'm going to let you take a moment to

take a closer look at that reduced compressability graph.

And maybe gain some appreciation for certain key points on it.

Assess yourself, and then we'll return. Alright well I just want to drive home

the key point one more time, associated with this particular piece of lecture

video, and that is the corresponding states.

They correspond when they are considered at conditions relative to their critical

condition. So, let's just look at a, a couple

specifics instead of trying to plot 25 different gases all on one plot.

So, shown here is the behavior of Ethane gas at 500 Kelvin which is a reduced

temperature of 1.6375. All right?

So, it is 1.6375 times higher in temperature, than the critical

temperature. So, could work out in a little calculator

if you like, what the critical temperature must be by dividing 500 by

that. But we won't worry about that for the

moment. I'll just show you that the experimental

data, those are the open circles. And also illustrate how two equations of

state are doing here. So, the Van der Waals equation, has the

right shape, but it dips down a little too much in the compressibility.

That's what's being plotted on the left. And we're plotting that against molar

volume. The Redlich–Kwong equation of state is

also shown. And it seems to do extremely well

actually for ethene over this temperature range.