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

Well, that last demonstration was somewhat more tame than some you might

have gotten used to, I suppose. I apologize we didn't take any pressures

up to explosive values, but hopefully it gave you a feel for the volidity of some

of the gas laws, that are related to ideal gas equation of state.

Now let's turn our attention to describing the behavior of gases over a

broader range of temperature, pressure, and molar volume.

So, I just remind you and catch you up where we left off.

We had looked at the behavior of certain gases, as they approach ideality.

So, here's our equation of state again, expressed in molar volume, noting that at

a constant temperature, all gases converge to ideal behavior, as the

pressure goes to 0. That is they all converge to a constant

value of pressure times molar volume, PV bar.

So PV bar equal RT, that really expresses that if the temperature is constant,

given the universal gas constant is a constant, then PV bar must be constant

for an ideal gas. And that does happen, but only at very

low pressures. So, let's actually look at the ratio of

PV bar over RT. That ratio is called the compressibility.

For an ideal gas, since PV bar equals bar RT, it should be equal to 1 under any set

of conditions. And so, I have a plot here, just like on

the last slide, of PV bar. But now, PV bar over RT, but at a

constant temperature. And in this case, the temperature is 300

kelvin versus the pressure. This pressure is however, expands over a

much larger scale than we were looking at previously.

So 1 bar is about 1 atmosphere, now we're actually looking at a pressure range that

goes up 1,000 atmospheres. Now, were the gas to be ideal, never the

less, the compressibility would be 1 at all pressures.

So, that defines an ideal gas, compressibility equal to 1.

However, what you notice is for three gases that are shown on this slide,

helium, methane and nitrogen, none of them are a horizontal line by any means.

And in fact, the behavior is sort of variable.

The methane line actually dips below, to a compressibility below 1, and then it

passes back through 1 and rises to a higher value.

And it looks as though nitrogen and helium, at least on the scale we can see,

it's not clear if they might dip down a, a little bit here at the lower pressures,

although they're still pretty high compared to what we're sitting in as an

atmosphere. nevertheless they all rise rather steeply

as we get to very high pressures. And if instead of focusing on comparing

different gases, we just take a single gas, and look at the change in

compressibility as a function of temperature.

So this is all methane data. And it's at 200, 300, 400, and 600

kelvin. And again, over a pretty large range of

pressures plotting the compressibility. So whenever you see z, just think PV bar

over RT, and it should be 1 for an ideal gas.

So, this point here, 1, the ideal gas, sure enough, that's what they all should

converge to at low enough pressure. So at 0 pressure, they'll all start

there. But, you see the behavior, again, is, is

relatively complex, so at the lowest temperature, 200.

It drops down and gives rise to this big bulge below the lines, you might say, and

then goes shooting up eventually. The bulge gets less and less deep, as the

temperature rises. Until by the time where it's 600 it

doesn't even look as though maybe it, it ever goes down at all.

But in any case there is this varied behavior.

Well, well let's think physically about what that means.

So if the compressibility is less than 1, one way to think about that is that the

molar volume is smaller than it would be for an ideal gas.

All right, if it were an ideal gas, the compressibility would be 1.

It's less than 1, so V bar for the real gas must be smaller than V bar for the

ideal gas. So, why would the amount of volume that's

taken up by a gas, be smaller? Well a good explanation for that, would

be to say that there are attractive forces operating between the gas

molecules that pull them together, cause them to occupy a smaller volume.

And so, evidently for methane, and as we'll see for many gases, at low

temperatures and low pressures, there are attractive forces that reduce the real

gas molar volume, compared to the ideal gas molar volume.

Now what about at the very high pressures?

Imagine that we're really squeezing a lot of molecules together.

And molecules, are matter. They do occupy a certain amount of

intrinsic space. They are nuclei and electrons, and if you

try to squeeze them together eventually they, bump into each other.

And they repel each other, you can't have two things occupy the same space at the

same time. So it's that repulsion which causes the

volume to be larger, than would be true for an ideal gas.

And so what happens if the real gas volume, required to hold that gas, is

larger than that for an ideal gas? Well then the compressibility must go to

be greater than 1. And sure enough, that's what we see at,

I'll go over here with the multiple gases.

That's what we see at very high pressures, that all of the gases have

compressibilities well above 1. And maybe the last thing to think about

is, what would happen at high temperature?

So, temperature can be thought of as being associated with the kinetic energy

of a gas. The more temperature is in there, the

faster those gas molecules are moving around.

In fact they're moving so quickly, that maybe they don't have time to notice as

they close to one another, they're a little bit attracted to each other.

because instead, they're bashing into one another, immediately afterwards like

billiard balls, and feeling that repulsion.

And so in fact as we go to higher and higher temperatures, focusing over here.

So here is a constant pressure, let's say 200 kelvin.

It's the lowest temperature where the attractive forces manage to function, to

reduce the compressibility the most. And then, as we increase the temperature,

that, compressibility goes up and up and up, as the repulsive forces become more

felt. So at either high temperature, or high

pressure, or both, it appears that repulsive forces dominate, and we get

that the real gas molar volume, is greater than the ideal gas molar volume.

Alright, with those concepts to think about, let's pause for a moment and I'll

give you a chance to essay one or two problems, that address these issues.

Now let's turn to a towering figure in the history of Thermodynamics.

And that would be Johannes Diderik Van Der Waals.

And that's probably va, fondervalls, if my Dutch were better.

and so here you see a picture of him on a stamp from the Netherlands.

He was a Dutch scientist. And on that stamp is an equation, which

is a wonderful thing about Europe, you find equations on stamps.

Maybe many other places around the world too.

It's a little more rare in the United States, sadly.

But, this isn't, of course, about geopolitics, it's a of course, about

thermodynamics. So let's investigate that equation, and

make it a little bit larger so you don't have to squint at the stamp.

So Van Der Waals spent a lot of time thinking about non-ideality.

He knew that the ideal gas was nonsense at most pressures, temperatures, that we

would ever want to work at. And he developed an equation of state,

designed to properly predict the interrelationship of pressure,

temperature and volume, over that range of states, and so you see here it here.

It is quantity, pressure plus a over V bar squared times quantity V bar minus b

is equal to RT. So you see, you still have this equal RT,

as in the ideal gas equation of state. But PV bar has gotten a bit more

complicated. And what are these constants a and b?

Well, they are just that, constants. They're associated with individual gases

and they are a measure, a is a measure of the interaction strength between the

gases, the attractive interaction strength.

And b is a measure of the molecular size. And so just to take a look at some of

these constants for instance in appropriate units for 1 being divided by

V bar squared, and 1 1 not just standing alone.

Helium, compared to ammonia, the a perimeter, which tells you something

about attractive forces between the gas molecules, very, very small for helium, a

noble gas, no permanent electrical moments, doesn't attract each, molecules

don't attract each other much. Versus ammonia which can actually

hydrogen bond to itself in the gas phase, for instance, and you see a much, much

larger a value. And then if we look at the sizes,

actually helium and ammonia, not that different in size, at least based on the

b parameter ammonia about, half again, as large as is observed for helium.

There are other equations of state, that have been developed to describe the

behavior of gases well into their non ideal ranges.

And so I've just shown you the Van der Waals.

Another popular one is the Redlich-Kwong equation of state.

And again, you see it expressed, I, I've rewritten the Van der Waals here to

isolate pressure on one side. This is just simple algebra, rearranging

the equation. In the Redlich-Kwong, written in the same

way, with pressure on the, on the left side.

You see all the terms we expect, the universal gas constant is in there, the

temperature is in there, the molar volume is in there.

The temperature appears again to the half power, so that's a little bit different

in this case. And there are two empirical constants

again, A and B. And so again, all gases will be

characterized by their own values of capital A and capital B.

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.

And I'm showing you the behavior of the Redlich Kwong and the Peng Robinson

equations of state. You see they don't do too badly.

I'm not showing the Van der Waal equation of state, because it actually does

bizarre things. And in fact, it predicts negative

pressures. And, why would that be?

So let me just rewrite the Van der Waal's equation of state.

If you look at it, I've got a term that, you know, in the limit of the density

going infinitely large. Remember that the density is 1 over the

molar volume. So here's 1 over the molar volume, that

would be like a term times density. Here's one over the molar volume squared,

that would be like a term times density squared.

A and B are positive constants, so ultimately rows squared, will dominate

over row. There will be some point where in the van

der Waals equation of state, pressure will become a negative number.

And it turns out for these conditions, that happens relatively early on, and you

just get nonsense. Well, okay with that to think about let's

spend a moment here and I'll let you answer a question about Van Der Waals

perimeter. Alright, well, I hope you've internalized

the details of these non ideal equations of state, cubic equations of state.

We're going to employ them in the next lecture, and in particular, we're

going to employ them as we look at, gas liquid, pressure volume diagrams, or PV

diagrams. [SOUND]

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