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Let's, return to the law of corresponding states now and look at it in a little bit
more detail. So, when last we, paused to have some fun
with the different phases of dry ice, carbon dioxide.
I showed you this, table of critical parameters, temperature, pressure, molar
volume for a variety of gases. And noted that although the individual
parameters, the critical temperature, or the critical pressure or the critical
molar volume, was really very substance dependent.
an enormous range from 5 Kelvin all the way up to 304 Kelvin for the gases on
this slide. Nevertheless, when you instead take a
property like the compressibility measured at the critical point, you get
near constant behavior. And I rationalized why the value might be
near where it is based on the Van der Waals expressions for these critical
properties. So, there's this correspondence between
different real gases independent to the equation of state.
So, let me actually define then, a reduced equation of state, a so-called
reduced equation of state. So, this is just the Van der Waals
equation of state, again showing what we derived the relationship to be.
Between Van der Waals parameters and critical parameters.
1:22
So, if now I rewrite the Van der Waals equation of state, changing its
variables, so instead of using pressure, I'm going to use pressure divided by
critical pressure. And I'll use molar volume divided by
critical molar volume. These are, of course, measurable
quantities, they're just numbers, the critical pressure and the critical molar
volume. Once I have measured them and I've
written them down in table somewhere, I'll just be dividing by some constant
value. But I'm going to give those variables a
name, I'm going to call it the reduced pressure, the reduced molar volume, and
there's also a reduced temperature. So, you can think of this as being the
quantity has a value of 1, these reduced quantities have values of 1 at the
critical point. And they'll either be above 1 or below 1,
if they exceed or are less than their respective critical values.
So, when I do that, when I rewrite the equation of state.
Not using a and b because by inserting, by doing these divisions by the critical
parameters, which involve a and b. All of the a and b terms drop out.
Entirely equivalently to the derivation in the last slide.
So, you end up with this universal equation for all gases, that the reduced
pressure, plus 3 divided by the reduced molar volume squared, times the quantity
reduced molar volume, minus the 3rd. Is equal to 8, 3rd, times the reduced
temperature. So, this is an example of the Law of
Corresponding States. It says that all gases will have the same
properties if compared at corresponding conditions, where corresponding
conditions really means relative to their critical conditions.
They'll all have different critical conditions, but when you look their
behavior relative to their critical conditions, they all behave the same.
3:20
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.
8:43
Now, let me show you a different gas, this is argon.
And this is not argon at 500 Kelvin as ethane was.
This is argon at 247 Kelvin, so only half the temperature.
It's quite a lot cooler, but you would burn yourself rather badly if you stuck
your hand into a 500 Kelvin gas whereas 247 Kelvin, that might feel good on a hot
summer day. And if we again look at the experimental
data, and the behavior of the equation's estate, they show sort of similar
behavior. Again the Redlich-Kwong is a somewhat
better equation of state than the Van der Waals.
But the key point I want to make is Argon's critical temperature is very
different than Ethane's critical temperature.
When I divide 500 by ethane's critical temperature, I get 1.6375.
When I divide 247 by argon's critical temperature, I get 1.6363, very close to
1.6375. That is, each of these two plots is for
the same reduced temperature, even though it's for a completely different actual
temperature. So, if I do indeed compare them at the
same critical temperature, and in this case I will do 1.64, if I round this to
only two digits they're right at the exact same temperature.
That's the beauty of rounding, I guess. I'm going to plot, now I'll actually plot
reduced pressure against compressibility just to have something to illustrate.
And you see that ethane and argon behave exactly equivalently, this is a good
example of the Law of Corresponding States.
But, the take home message was, they're not at the same temperature, they're at
the same reduce temperature. Alright.
Well we have, explored several equations of state now.
We've mostly focused on the Van der Waals.
Although we've looked at Redlich-Kwong and Pang Robinson.
In the next lecture, I want to take a look at yet another equation of state.
And one that has properties that make it especially useful particularly
appropriate within the room a statistical mechanics and that is the the Virial
equation of state that will come next