So, to return to our original theme, why are there "phase transitions" (the general term for phenomena like melting and evaporation, where substances have jumps in properties) in fluids? We just spent a lot of time, as you no doubt have done in class, talking about how molecules bouncing around are really pretty simple. I mean, as pressure goes up, volume goes down, and vice versa. So what?

The catch is that there was a special condition we didn't mention. We were assuming that the molecules don't interact with each other, only with the walls of their container. Boyle's Law is more properly stated as: at constant temperature, in a thin gas, the pressure multiplied by volume is a constant

or as an algebra formula, PV = constant

The point of the "thin" gas is that the molecules should be on average pretty far apart so that they don't collide with each other very often. This caveat of course immediately leads to the question of what happens if we're not talking thin gas? How does the P vs V curve change if we look at the low-volume, high-pressure part where molecular interactions are significant because the molecules are crammed together? (For air, this actually has to be fairly high pressure by ordinary human standards, much higher than atmospheric pressure) This regime, as physicists say, must be where the mysterious gas-liquid explanation lurks.

Return with us now to those days of not-quite-so-yesteryear, two centuries after Boyle, to the late 19th century, when Johannes Diderik van der Waals is developing the ideas that will win him one of the first Nobel Prizes. (He wasn't anticipating it, however, since Alfred Nobel hadn't died yet...the first Nobel Prize was in 1901) Here he is, both as the young man and as the great professor looking rather severe, like the usual 19th century portrait. As usual, most pictures are of the older scientist after they've become famous: here's a lab photo with a student, Kamerlingh Onnes, who received a Nobel three years later. Can you name some other important Dutch physicists?

Van der Waals used the picture of the inter-molecular force law we looked at above. He saw that with two basic numerical values you could describe the force law pretty well:

• one value was the diameter of the "hard core" of repulsion (analogous to the size of a billiard ball) used in a very simple but clever way: the core reduces the effective volume available for the other molecules to move in. The V in Boyle's Law is replaced by (V-b) where b is the total volume of the hard cores of the molecules. This nicely works out to Boyle's Law as a limiting case: when the gas is thin, V is much larger than b, so you can neglect b and not be far off. (In fact, we've quietly done this already in the previous page: if you go back and look carefully at the simulation, you'll notice the scale begins slightly off the bottom of the chamber, by the width of a molecule. This is because we've made the molecules so big to be visible)
• the other value was another clever trick to deal with the inter-molecular attraction: its physical effect is to increase the effective pressure. The P in Boyle's Law is replaced by (P + a/Vsquared) where a is a constant measuring the strength of the attraction. Again for large V the correction term can be neglected and Boyle's Law rules.

Now, this may look to you like the Equation From Hell, but it's actually not bad, as the people who swim outside while there's still snow on the ground say. We'll drop the algebra now and do everything with graphs, so you can still say you know the van der Waals Equation. The picture of how it departs from Boyle's Law is all you need to understand.

Here's a graph of what it looks like for some physically reasonable choices of the two parameters: If your browser supports Java, you can change the values of the parameters and see the curve change shape. If you set both parameters to zero (the left) you get the Boyle's Law curve (a hyperbola). If you then leave the attraction zero and increase the size of the repulsive core, the curve stays the same shape but moves to the right. This makes sense if you think about balls of a certain size being packed into smaller and smaller volume: the maximum is reached when all the balls touch, which depends on the size of the balls. Bigger balls will reach the maximum at bigger volumes.

The attraction strength parameter is a little less obvious: as it increases, it cuts a trench out of the curve. This is because molecules that spend a lot of time close together because of attraction will not be bouncing off the walls as often, and so there will be less pressure measured. This gets stronger (more pressure reduction) as the volume is decreased until the volume becomes so small that the molecules are jammed together and the repulsive core drives the pressure up again very fast.

You'll notice that the shape of the curve depends on just how strong the two parameters are. This indicates how the properties of the substance depend on the exact numerical values of those intermolecular forces for a particular substance (and temperature). We're not showing any numbers here, to keep things simple, but you should get a feel for this dependence.

What's significant for our purposes is a very simple observation: with the starting parameters,

There's a bump on the curve

That's the key! It means that for any one value of pressure, there's more than one possible value of volume. For example, look at the black line in the graph. For that pressure, there are three possible values of volume, at points labeled A, X and B. This just follows from the math (it's a cubic polynomial, so it has at most three solutions), but what it means physically is that there may be more than one configuration of molecules at this pressure (and temperature...remember that this is all being done at a constant temperature).

You can guess what they might be, but first let's see what we can learn about their properties just from the curve.

• The fluid at A is the same number of molecules in a much smaller volume than at B, many times smaller, so A-fluid is many times denser than B-fluid.
• The fluid at B is sensitive to changes in pressure: a small pressure change makes a big volume change (the slope of the curve at B is shallow) while the fluid at A is just the reverse: a big pressure change makes little difference to the volume. So A-fluid is relatively incompressible while B-fluid is pretty compressible.

Sure enough, this is what we expect for gas and liquid. Liquid exists at the low volume part of the curve, and gas exists at the high volume part.

We've not mentioned the third point on the curve, the middle one X. Is this some other state of matter, intermediate between gas and liquid, that the Government is holding in Roswell NM and doesn't want you to know about? Alas no, it turns out not to exist as a real state, but for interesting reasons.

Fair warning: this gets a little heavier going.

It doesn't exist because any molecules in the middle part of that curve are in an unstable arrangement. The reason is the slope of the curve: we'll walk through what would happen if you somehow managed to make a lot of molecules in that state. Imagine it exists under a piston exerting the pressure P, as in the previous page. Since all the properties like pressure are statistical averages over the many molecules bouncing off the walls, the pressure actually fluctuates slightly around the average P from instant to instant. Normally that's not noticed because the fluid is stable: any fluctuations generate forces to restore the previous condition.

The usual analogy is mechanical: consider an apple in a bowl, versus a pencil balanced (for an instant) on its sharpened tip. Any fluctuations (such as slight vibrations in the table, air currents, worms, whatever) will cause the pencil to fall over, but the apple will stay where it is. The reason:

• Any change in position of the pencil causes forces will make the change bigger: once the pencil is tipped, its weight is off center and it starts to fall further in the same direction, building up more force and speed as it goes. This is what is meant by unstable.
• The situation is exactly reversed for the apple: if it is slightly moved by a vibration, it can only move uphill in the bowl, away from the bottom center. Thus it then feels a force that pushes it back (downhill) to where it was. It is thus stable.

Now back to the fluid (remember it?). The fluid at both A and B is stable, because of the nature of the restoring forces after a fluctuation. Suppose the pressure drops for an instant in the fluid (fewer molecules happen to bounce off the top in that instant). The weight is now greater than the pressure, so the top plate starts to drop, reducing the volume slightly. What happens to the pressure inside? If we look at the P vs V curve at A or B, we see that as the volume decreases the pressure increases Physically, this means the pressure is now greater than the weight, so the top lid moves up, restoring its previous position.

Walk thru that mentally again.

Also, walk thru yourself what happens if the initial pressure fluctuates up because a few more molecules hit the lid in an instant. Really, do it!

The situation of the fluid at A and B is thus like the apple's situation: it is stable because it feels restoring forces after any fluctuation.

Now consider the fluid at point X. What happens if the pressure drops for an instant and the volume decreases slightly? The pressure decreases more! (look at the curve at point X) The weight keeps dropping, the volume gets smaller and smaller, and the pressure gets smaller and smaller! It's a pencil falling over!

The result is that fluid in condition X cannot last, any more than the pencil can stay poised on its tip. Fluid at X is unstable, and will convert to something else a lot faster than the pencil will fall, so X-fluid will never exist in the real world.

One takeaway from this is a little lesson on extracting physical meaning from graphs: on a P vs V curve, a slope \ is a stable region and a slope / is unstable.

Which is why there are can be only two extreme conditions for this fluid at this temperature, gas and liquid, with a gap in the middle where nothing exists. Well, nothing uniform ... actually what you get is a mix of liquid and gas.

That's not all, Folks!

This curve is part of what's called the Equation of State, which is extreeemely useful in understanding all sorts of properties of substances. It includes the effects of temperature, which we've held constant in our discussion. One of the things van der Waals explained is how you can change a substance from liquid to gas without going through a jump. Modeling the equation of state for everything from the interior of stars to the interior of chemical plants is active research, because it is so useful in understanding how stuff behaves. Van der Walls' Equation was only a start. It explains the major features of what's going on in a fluid, including a lot of features we didn't mention here like "critical points", but (alas) getting accurate agreement with real measurements on real fluids requires more complicated equations.

"There is no problem, no matter how big and complicated, which when looked at in the right way, does not become still bigger and more complicated."

What real physicists currently think about States of Matter...not for the timid, but the beginning is interesting to skim.

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