This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Statistical Molecular Thermodynamics

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This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Module 2

This module begins our acquaintance with gases, and especially the concept of an "equation of state," which expresses a mathematical relationship between the pressure, volume, temperature, and number of particles for a given gas. We will consider the ideal, van der Waals, and virial equations of state, as well as others. The use of equations of state to predict liquid-vapor diagrams for real gases will be discussed, as will the commonality of real gas behaviors when subject to corresponding state conditions. We will finish by examining how interparticle interactions in real gases, which are by definition not present in ideal gases, lead to variations in gas properties and behavior. Homework problems will provide you the opportunity to demonstrate mastery in the application of the above concepts.

- Dr. Christopher J. CramerDistinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics

Chemistry

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.

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.

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.

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

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