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

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From the course by University of Minnesota

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.

From the lesson

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 take a look now at another equation of state.

The virial equation of state. So, what is a virial expansion that plays

a role in these equation of state? The virial expansion expresses the

compressibility. And you'll remember that the

compressibility is P V bar, divided by RT.

As an infinite series expansion in either the density and remember that the density

is the inverse of the molar volume. So I could either write this as an

expansion in density or an expansion in inverse molar volume, or finally we can

use the pressure. So expressed in inverse molar volume the

expansion is as shown here. It's 1 plus B2VT over V bar.

Plus B3VT over V bar squared and a term over V bar cubed, and so on.

Infinite series. If we express, instead, in the pressure,

1 plus B2PT times the first power of the pressure, plus B3PT times pressure

squared, pressure cubed, and so on, an infinite series expansion.

The constant terms that are appearing, the B terms are known as Virial

coefficients. They are indexed by two subscripts, the

first subscript tells us at what term in the expansion do they appear, the second,

the third, the fourth. The second subscript is simply are the

volume term or a pressure term. And so, for instance, we would call B2V a

second virial coefficient. They are functions of temperature, and

that's emphasized by including temperature in parentheses afterwards.

So if and when we determine them, we'll do them at certain temperatures.

And we'll get to that momentarily. Do, take a moment to look at the

equations and think about sort of the sensibility at certain limits, and in

particular, let's consider the limit as pressure is going to 0.

So as the pressure is going to 0, and as my molar volume as a result of there

being essentially no pressure is going to infinity.

And if the molar volume is going to infinity, then, of course, the density is

going to zero because they're inversely related.

Well, what would happen to these equations?

If pressure is going to zero, pressure cubed goes to zero really quickly.

All the terms are going to zero, including even this one.

Although most slowly, this one. And ultimately, I'll get compressibility

equal to one. Which is what I expect.

I should have ideal gas behavior as my pressure goes to zero.

Similarly, if molar volume is going to infinity, I keep dividing things by

infinity or powers of infinity, they all go away, and I'm left with

compressibility equals one. So the virial equation of state has the

correct form to show ideal gas behavior at that limit of infinitely low pressure.

Well, where do these virial coefficients come from?

So, let me show you a plot of compressibility against pressure.

Very near no pressure at all, so these are [COUGH] very low pressures, somewhere

between zero and a tenth of a bar. [COUGH] And remember that a tenth of a

bar is about a tenth of an atmosphere. And what we see for these isotherms.

And this is for ammonia gas. Is zero degrees C, 100 degrees C, 200

degrees C. If I extrapolate, they're all going to

ideal gas behavior at the lowest pressure.

More over they're all linear. And let me think about what happens at

very low pressure. So as the pressure is going to 0, as

numbers go to 0 if you square or cube or take their fourth power they go to 0 much

faster obviously. So the only term that will survive that

contains P within it. In the virial expansion, it will be Z is

equal to 1 plus B2P, function of temperature, divided, sorry, multiplied

times P. So that is a linear equation, in P.

And sure enough, these are lines, as a function of P.

And so the slope, of this experimental measurement, gives you, B2P.

And each of these isotherms, has a different slope.

And that illustrates to you that yes the coefficient does depend on temperature.

But, by doing a series of measurements, you can infact determine B2PT, for a

given gas at given tempuratures. You can also manipulate the two virial

expansions. It takes about maybe 15 lines of

equations to do this. It's not awful, but it's a little much to

put on a slide, so I'll just give you the result.

You can show that B2V(T), so that's the coefficient that appears in the molar

volume, virial expansion, Is equal to something quite simple.

It's just R times T times B2PT. So by doing this experiment, you

simultaneously get both of these two virial coefficients, second virial

coefficients. Let's just look at, how the various terms

in the virial expansion. What sort of orders of magnitude they

take on,at, under different sets of conditions.

So here are some data for argon at room tempature, 298 Kelvin.

I'll show you for one bar, so that's atmospheric pressure, 10 bar, and 100

bar. And these all are the B 2V component of

that stage, as in units of volume per mole.

'Cuz remember compressibility is unitless.

1 is certainly unitless. So, if we're going to divide by Vbar,

B2VT has to have the same units as Vbar, in order to be unitless.

What are the units on Vbar? Volume per mole.

In any case, at one bar, argon, it really behaves pretty ideally, so, the one comes

in. And the second term in the virial

expansion is 0 point 00064. So the deviation from the ideal behavior

is 6 times 10 to the minus 4th. Pretty small.

And the third and fourth terms don't contribute at all.

On the other hand as I go up to 10 bars, so I increase the pressure by a factor of

10. And it turns out each of the contribute,

well, okay, the one contribution that wasn't zero before, also goes up by about

an order of magnitude. So it went from 6.4 times 10 to the minus

4th, to 6.5 times 10 to the minus 3rd. And the 3rd virial term begins to creep

up a bit. It's actually non-negligible.

There's a little bit a contribution from all the other terms, but it's still

pretty far out past the decimal place. If I increase the pressure yet again by

an order of magnitude, now the second virial term is up by about a factor of

ten. The third virial term is up by about a

factor of ten. And there's more a contribution from the

final terms. So, you can sort of see that the, the

length of the expansion you need to consider increases as you move further

and further away from ideal gas length conditions.

So, okay it's an equation of state. It's got some constants appearing in it.

Can we gain any physical insight from these constants?

That is what does B2V, what does it mean? Well it actually is possible to assign a

pretty straight forward and simple meaning to B2v.

So let me just write again the low pressure virial expansion in pressure.

It says that the compressibility is 1 plus B2P, times pressure.

And the other terms drop out because the pressure is low enough they don't

contribute. Well, when we multiply both sides by RT

over P. So that'll leave only the molar volume on

the left-hand side, one times RT over P gives RT over P.

And the Ps will cancel out over here. I get RT times B2P.

But, remember, RT over P. What is that?

That's Vbar for an ideal gas. RTB2P, what is that?

That's B2V. We, showed that on a prior slide, or I

told you that you could actually derive that relationship.

So, I'll just replace those in there. It says Vbar for my real gas is Vbar for

an ideal gas plus B2V. And all I have to do is rearrange that

ever so slightly. It says B2V is Vbar for the real gas

minus Vbar for an ideal gas. That is, B2V is the difference between

the observed molar volume and the ideal gas molar volume for a given set of

pressure and temperature conditions. And that's a pretty intuitive phenomenon,

and intuitive quantity, if you will. So, let me take a moment and, and see if

that's that is intuitive for you. I'll let you answer a question that will

address that in a little bit more detail. Well, now that we have this intuitive

feeling for what B2V means. Let's actually take a look at some B2V

data as a function of temperature for a variety of gases.

So I've got nitrogen gas, methane, carbon dioxide on this slide.

And I am plotting b2v versus temperature. And remember that the physical

interpretation of b2v is it's the deviation of the real volume from the

ideal gas volume. So here, this line tracking across the

data at, at zero. That would be if the observed volume was

indeed the ideal gas volume. Right?

And, in that case, B2V would be zero. And so you see, as for helium for

example. As we start very, very cold, the B2V is a

negative number, so the actual molar volume is smaller than the ideal gas

volume. The gas molecules are clumping together

would be a way to think about that. They're occupying less volume than an

ideal gas would. And then as I raise the temperature, I go

up, up, up, I pass through the ideal gas volume, and I go a bit above, and then I

kind of level out after that. And then helium hovers pretty close to

the ideal gas volume at higher temperatures.

Nitrogen, methane, carbon dioxide, they all show the same behavior.

That at lower temperature, they are occupying less volume then an ideal gas.

They are sticking together for lack of a better way to think about it.

As you warm them, they all increase, and they all do ultimately pass through on

this scale. We don't see Co2 pass through here, but

clearly they're all approaching that ideal line ultimately.

And so again what we're seeing is that attractive forces are dominating in the

low temperature region. That all of these gases are occupying

less molar volume, and we're seeing that expressed as B2V.

And ultimately we get repulsive at high temperature, they all pass through, once

you're on this side of the line. Your molar volume, because you've got a

positive V to V, is larger than the ideal gas volume.

So they must be repelling each other and be forced to occupy more space.

All right, well, we've invoked, a physical phenomenon, or really two.

We've invoked attraction and repulsion to explain changes in molar volume.

So it's really appropriate at this point that we spend a little time thinking

about the physics behind attraction and repulsion.

And so in the next video, we will take a look at molecular interactions.

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