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

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

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

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

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

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

Â broader range of temperature, pressure, and molar volume.

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

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

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

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

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

Â value of pressure times molar volume, PV bar.

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

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

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

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

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

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

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

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

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

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

Â much larger scale than we were looking at previously.

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

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

Â less, the compressibility would be 1 at all pressures.

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

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

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

Â And in fact, the behavior is sort of variable.

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

Â passes back through 1 and rises to a higher value.

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

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

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

Â atmosphere. nevertheless they all rise rather steeply

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

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

Â compressibility as a function of temperature.

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

Â kelvin. And again, over a pretty large range of

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

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

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

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

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

Â relatively complex, so at the lowest temperature, 200.

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

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

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

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

Â But in any case there is this varied behavior.

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

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

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

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

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

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

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

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

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

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

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

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

Â Now what about at the very high pressures?

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

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

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

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

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

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

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

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

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

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

Â I'll go over here with the multiple gases.

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

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

Â is, what would happen at high temperature?

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

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

Â faster those gas molecules are moving around.

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

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

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

Â billiard balls, and feeling that repulsion.

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

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

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

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

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

Â felt. So at either high temperature, or high

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

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

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

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

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

Â And that would be Johannes Diderik Van Der Waals.

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

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

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

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

Â Maybe many other places around the world too.

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

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

Â thermodynamics. So let's investigate that equation, and

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

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

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

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

Â designed to properly predict the interrelationship of pressure,

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

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

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

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

Â complicated. And what are these constants a and b?

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

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

Â gases, the attractive interaction strength.

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

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

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

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

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

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

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

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

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

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

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

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

Â behavior of gases well into their non ideal ranges.

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

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

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

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

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

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

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

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

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

Â in this case. And there are two empirical constants

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

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

Â And yet another equation, is the Peng-Robinson equation of state.

Â Once more, a somewhat complex relationship more complex perhaps than

Â the Van Der Waals, and the relevant parameters are not called A and B

Â anymore, they're called alpha and beta. But in any case, they serve a similar

Â function. They are designed to say something about

Â attractiveness or repulsiveness of the molecules in the gas phase.

Â And I've called these cubic equations of state, in the title of the slide, because

Â if we were in fact, to expand these out, take these V bars out of the denominator,

Â and I'll show you examples of that in not too long, we would end up with an

Â equation that is cubic in V bar. And that has key implications, and let's

Â explore that momentarily. But first let me just actually show you

Â some of this behavior. And so on this graph I've go pressure on

Â the left hand side on the Y-axis, and density on the X-axis.

Â So we haven't seen density before, but let me just note for you, density is just

Â the inverse of molar volume. And so that's sort of obvious if you

Â think about the units. Density is mols per liter, for example.

Â And that, and molar volume is liters per mol.

Â So they're just related to one another as the inverse.

Â And you can write equations either way, if you like to.

Â and this is just something to keep in mind, because we may use either of these

Â units in the future. A liter is a cubic decimeter, so

Â sometimes thermodynamics prefer decimeters to liters.

Â In any case these are data for ethane as a gas at 400 k, and shown here are the

Â experimental data for the change in density, as a function in the change in

Â pressure. And so what you see is that, as the

Â density is increasing, that is as you're compressing the gas, you're putting more

Â mass into a smaller volume. The pressure is going up, you must

Â squeeze on the gas to force it into that volume.

Â So the experiment is the solid line here, and that does not mean solid phase

Â ethane, it's just the solid line on the graph.

Â you see the, the Van Der Waals equation of state, does well up to about a density

Â of maybe, 7 it looks like. And then it starts to predict that much

Â higher pressures would be required, to keep compressing that are observed.

Â On the other hand, Redlich-Kwong and Peng-Robinson are doing reasonably well.

Â And so those somewhat more complex equations of state, have improved

Â performance at very very high pressures. So let's actually cool our ethane down a

Â little bit. Let's take it does to 305.33 degrees

Â kelvin. At that temperature, we observe something

Â happen, as we keep pressing on a piston. So if you imagine that you've got a

Â volume in here of gas, and I'm decreasing that volume by pressing on a piston.

Â At a certain point, I would observe that the pressure I'm applying, I don't have

Â to increase pressure but the piston keeps going down, and if I look inside my

Â container I would discover, that's because my gas is liquifying.

Â I'm seeing a liquid phase appear at the bottom of the container.

Â And that will continue, until the piston finally touches the top of the liquid,

Â and now suddenly I'm going to have to increase the pressure dramatically to

Â compress it any further. So, that's what's really being shown

Â here, the experiment is again the solid line.

Â I've increased pressure up, up, up. And then suddenly I hit this flat part,

Â where keeping the pressure constant, the density just keeps getting larger and

Â larger, and that's the average density. because the density of a liquid, is much

Â larger than the density of a typical gas. And so, it'll transform itself from gas,

Â all the way over to liquid, with its normal liquid density, at that pressure

Â and temperature. And then if I want to keep compressing a

Â liquid, that's much harder than compressing a gas, and the pressure

Â shoots up. So we say that the liquid and the vapor

Â are in equilibrium over this range. They're both present, and until I go

Â beyond a certain pressure, they will both be present.

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

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

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

Â bizarre things. And in fact, it predicts negative

Â pressures. And, why would that be?

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

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

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

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

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

Â that would be like a term times density squared.

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

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

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

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

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

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

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

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

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

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

Â diagrams. [SOUND]

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